A Remark on the Puig Correspondence and Thévenaz’s Lifting Theorem

Hubert Fottner

Friedrich-List-Str. 7, 86153 Augsburg, Germany

In the authors joint paper with Burkhard Külshammer [6] it is shown that Thévenaz’s lifting theorem [14] is a direct consequence of the following general ring theoretical fact:

Lemma A ([6] Proposition 1.1). Let R be a ring, denote by J(R) its Jacobson radical, and let e_{1}, …, e_{n} be idempotents in R such that e_{i}e_{j} ∈ J(R) for all i ≠ j and e_{1}+…+e_{n} ≡ 1 (mod J(R)). Then u := e_{1} + … + e_{n} is a unit in R and f_{1} := u^{−1}e_{1}, …, f_{n} := u^{−1}e_{n} are pairwise orthogonal idempotents in R such that f_{1} + … + f_{n} = 1.

We refer to [6] for a (rather trivial) demonstration of how lemma A applied to G-algebra theory yields a generalization of Thévenaz’s lifting theorem. However, let us emphasize also in this paper that this G-algebra application of lemma A is roughly equivalent to a result which was first proved by Fan Yun. So far unpublished, Fan’s proof recently appeared in a joint paper by Y. Fan and B. Külshammer [2].

The objective of this paper is to show that lemma A can also be applied to the Puig correspondence [12], [13], which allows to prove a lifting theorem à la Thévenaz in this context. As an immediate consequence, we obtain that the Green correspondence [8] commutes with induction in the sense of theorem C, below.

Throughout, we work entirely within Puig’s theory of pointed groups [12], [13]. The reader is refered to [10] or [15] for text book accounts of the theory. A brief introduction to the notation and the results used herein is given in section 1.

Let us fix a complete discrete valuation ring O of characteristic 0 with algebraically closed residue field 𝔽 of prime characteristic p. Unless stated otherwise, all our modules and algebras are finitely generated over O. But we do not exclude the possibility of non-trivial O-torsion (cf. section 1). Moreover, algebras are associative and have a multiplicative identity, and module means unitary left module.

Before we are able to state the main result of this paper, we need the following terminology:

Defintion B. Let G be a finite group, and let H_{α} and K_{β} be pointed groups on a G-algebra A. Then H_{α} is called relatively free to K_{β} (or relatively K_{β}-free, for short) if K is a subgroup of H, there are e ∈ α and f ∈ β such that f(^{h}f) = 0 for all h ∈ H ∖ K, and e = Tr_{K}^{H}(f). Moreover, H_{α} is said to be relatively free to K (or relatively K-free) if K is a subgroup of H, and there exists a point β of A^{K} such that H_{α} is relatively K_{β}-free.

In the situation of definition B, assume that H_{α} is relatively K_{β}-free. Then H_{α} is clearly relatively K_{β}-projective. Apart from that, K_{β} ≤ H_{α} as plainly ef = f = fe (if e and f are chosen as in definition B).

The notion for pointed groups of being relatively free is adapted from [2] and replaced our original manner of speaking in an earlier version of this paper where, in the situation of definition B, H_{α} is said to be “induced from” K_{β}. The following fact explains the terminology: Assume that A = End_{O}(M) is the G-algebra associated with an OG-module M. Then H_{α} is relatively K_{β}-free if and only if K ⊆ H and for some (or, equivalently all) i ∈ α and j ∈ β, we have

i(M) ≅ Ind_{K}^{H} j(M)

(where Ind_{K}^{H} j(M) denotes the induced OH-module).

Let G be a finite group. Denote by

G = γ_{1}(G) ⊇ γ_{2}(G) ⊇ γ_{3}(G) ⊇ …

the lower central series of G. Thus, for i = 1, 2, …, γ_{i+1}(G) := [γ_{i}(G), G]. Set γ_{∞}(G) := ⋂_{i∈ℕ} γ_{i}(G). Then γ_{∞}(G) is the unique minimal normal subgroup of G such that the corresponding quotient group of G is nilpotent.

Let H be a subgroup of G. Then γ_{i}(H) ⊆ γ_{i}(G) for all i ∈ ℕ, and so γ_{∞}(H) ⊆ γ_{∞}(G). Recall also that γ_{∞}(G) is a fully invariant, and in particular characteristic, subgroup of G.

Then main objective of this paper is to prove the following theorem.

Theorem C. Let G be a finite group, and assume we have the following inclusions of pointed groups on a G-algebra A

P_{γ} ≤ N_{ν} ≤ G_{α},
P_{γ} ≤ M_{μ} ≤ H_{β},

where P_{γ} is a defect of G_{α}, N = N_{G}(P_{γ}) and M = N_{H}(P_{γ}). Then the following assertions hold:

If G_{α} is relatively H_{β}-free then N_{ν} is relatively M_{μ}-free. If moreover there is a normal subgroup U of G containing P such that γ_{∞}(U) ⊆ H then the converse holds, as well.

We adopt the notation of theorem C. Note first that the additional hypothesis of the converse implication of theorem C is in particular satisfied if G = N_{G}(P)H; for in this case

U := ⋂_{g∈G}^{g}H = ⋂_{gH∈G/H} ^{g}H

is a normal subgroup of G which contains P as P ⊆ H and as there is a transversal for G/H consisting only of elements of N_{G}(P), by assumption. Moreover, we certainly have U ⊆ H.

Note also that, in the situation of theorem C, P_{γ} is a maximal local pointed subgroup (i.e. a defect) of N_{ν} as P_{γ} is a maximal local pointed subgroup of G_{α}. So N_{ν} is the Green correspondent of G_{α} (with respect to P_{γ}), and ν is the unique point of A^{N} fulfilling the given inclusion. Moreover, in any case, P_{γ} is a defect of both, M_{μ} and H_{β}, as well, and M_{μ} is the Green correspondent of H_{β} (with respect to P_{γ}); for if G_{α} is relatively H_{β}-free, we have in particular

P_{γ} ≤ M_{μ} ≤ H_{β} ≤ G_{α};

hence, by the above argument, P_{γ} is a defect of M_{μ} and of H_{β}, and thus M_{μ} is the Green correspondent of H_{β} (with respect to P_{γ}).

Conversely, in case N_{ν} is relatively M_{μ}-free, we have P_{γ} ≤ M_{μ} ≤ N_{ν}, and so P_{γ} is a defect of M_{μ}. Thus, by the Burry-Carlson-Puig theorem (cf. [15] theorem (20.4)), P_{γ} is also a defect of H_{β}, and M_{μ} is the Green correspondent of H_{β} (with respect to P_{γ}). In particular, given either hypothesis, μ is the unique point of A^{M} fulfilling the given inclusion.

Finally, note that if A = End_{O}(L) is the G-algebra associated with an OG-module L and j ∈ γ, i ∈ ν and e ∈ α then e(L) is an indecomposable OG-module with vertex P and P -source j(M), i(L) is an indecomposable ON-module with vertex P and P -source j(L), and i(L) is the Green correspondent of e(L) (with respect to P and j(L)). Of course, a similar assertion holds for the inclusion of pointed groups P_{γ} ≤ M_{μ} ≤ H_{β}.

Given the additional hypothesis of the converse implication of theorem C, one can apply the result twice, and obtains that an analogous assertion does not only hold for the Green correspondence with respect to the normalizer of a defect, but also with respect to an arbitrary subgroup containing this normalizer.

Corollary D. Let G be a finite group, and assume we have the following inclusions of pointed groups on a G-algebra A

P_{γ} ≤ K_{κ} ≤ G_{α},
P_{γ} ≤ L_{λ} ≤ H_{β},

where P_{γ} is a defect of G_{α}, N_{G}(P_{γ}) ⊆ K, and N_{H}(P_{γ}) ⊆ L ⊆ K. Then the following assertions hold:

(a) Assume that V is a normal subgroup of K containing P such that γ_{∞}(V) ⊆ L. Then K_{κ} is relatively L_{λ}-free if G_{α} is relatively H_{β}-free.

(b) Assume that U is a normal subgroup of G containing P such that γ_{∞}(U) ⊆ H. Then, conversely, G_{α} is relatively H_{β}-free if K_{κ} is relatively L_{λ}-free.

Adopt the notation of corollary D. Note that in case K is contained in N_{G}(P), the normalizer of P in G, V := P certainly satisfies the addtional hypothesis of assertion (a). So that corollary D is in fact equivalent to theorem C. We will prove corollary D directly (without proving theorem C first) using the fact that a result analogous to theorem C holds if the Green correspondence is replaced by the Puig correspondence (cf. theorem (2.6)).

Assume that P ⊆ U ⊴ G and γ_{∞}(U) ⊆ H as in assertion (b). Moreover, assume that H ∩ K ⊆ L. Then K ∩ U is a normal subgroup of K containing P. Besides,

γ_{∞}(K ∩ U) ⊆ γ_{∞}(U) ∩ K ⊆ H ∩ K ⊆ L,

by assumption. So that also the additional hypothesis of assertion (a) is satisfied. Therefore, in this case, G_{α} is relatively H_{β}-free if and only if K_{κ} is relatively L_{λ}-free.

Besides, note that, as by the observations subsequent to theorem C, one shows that, given either that G_{α} is relatively H_{β}-free or that K_{κ} is relatively L_{λ}-free, P_{γ} is a defect of L_{λ}, K_{κ}, H_{β} and G_{α}, as well, K_{κ} is the Green correspondent of G_{α} (with respect to P_{γ}), and L_{λ} is the Green correspondent of H_{β} (with respect to P_{γ}).

Recall that by Puig’s G-algebra version [11] (cf. also [15] corollary (23.3)) of Green’s indecomposability theorem [7], any pointed group G_{α} on a G-algebra A is relatively O^{p}(G)-free provided that G_{α} is relatively O^{p}(G)-projective. Applying theorem C to this result we obtain

Corollary E. Let U be a normal subgroup of a finite group G, and assume we have the following inclusion of pointed groups on a G-algebra A

P_{γ} ≤ H_{β} ≤ G_{α},

where P_{γ} is a defect of G_{α}, P ⊆ U, and γ_{∞}(U)P O^{p}(N_{G}(P_{γ})) ⊆ H. Then G_{α} is relatively H_{β}-free.

Note that, in the situation of corollary E, γ_{∞}(U)P is a subgroup of U as γ_{∞}(U) ⊴ G, which is normalized by N_{G}(P) as γ_{∞}(U) is characteristic in U, and hence normal in G. Thus γ_{∞}(U)P O^{p}(N_{G}(P_{γ})) is a subgroup of G.

A further application of corollary D is the following result:

Corollary F. Let P be a Sylow p-subgroup of a finite group G, let G_{α} be a pointed group on a G-algebra A with defect group Q such that Q is a normal subgroup of P and such that N_{G}(Q) is p-solvable. Moreover, let H be a subgroup of G, containing Q, the index of which in G is a power of p. Then G_{α} is relatively H-free.

Note that corollary F applies in particular if G is a p-group and Q = 1. In this case the result yields that G_{α} is relatively 1-free. So that corollary F can be regarded as an extension of Green’s theorem (in it’s G-algebra version). More generally, if Q = 1 and G is p-solvable then corollary F asserts that G_{α} is relatively H-free, where H is a Hall p′-subgroup of G. This special case of corollary F is [6] theorem 3.1, which in turn can be regarded as a common generalization of Green’s theorem and a classical result of Fong [4], and which is proved in [6] on the grounds of lemma A, as well. Using corollary D, we will reduce the general assertion of corollary F to this special instance.

1. Preliminaries and Notation

Let us briefly recall the notions of points, pointed groups and defects of such due to Lluis Puig [12], [13]. We refer to [10] or [15] for text book accounts of the theory.

Note first that, by our convention, any algebra A is semiperfect, that is idempotents can be lifted and A/J(A) is an Artinian ring, where J(A) denotes the Jacobson radical of A, even though A may contain some non-trivial O-torsion. (Cf. [1], for details.) So that also the Krull-Scmidt theorem holds for (finitely generated) A-modules. As J(O)A ⊆ J(A), A/J(A) is even a finite dimensional semisimple algebra over 𝔽, which is split, i.e. a finite direct product of full matrix algebras over 𝔽, as 𝔽 is algebraically closed. We leave it to the reader to verify that all the results of [15] we cite herein carry over to the more general setup of O-algebras which are finitely generated as modules over O, but not necessarily O-torsion-free (resp. O-free). [1] sections 42 up to 45 contain all the facts one needs.

(1.1) Let A be an algebra. Recall that U(A), the group of units of A, acts on the set of primitive idempotents of A by conjugation. The corresponding orbits (i.e. the U(A)- conjugacy classes of primitive idempotents of A) are called the points of A. Denoty by P(A) the set of points of A. Let α be a point of A. Then there is a uniquely determined maximal ideal 𝔪_{α} of A such that α ⊈ 𝔪_{α} (or, equivalently, e ∉ 𝔪_{α} for some e ∈ α). The assignment α ↦ 𝔪_{α} induces a bijection between the set of points of A and the set of maximal ideals of A. So that, in particular, P(A) is a finite set.

Let e be an idempotent in A. Recall that the canonical inclusion eAe ↪ A induces an injective map P(eAe) → P(A) (cf. [15] proposition (4.12)). (The homomorphism eAe ↪ A is the prototype of what is called a direct embedding of algebras.)

(1.2) Now let G be a finite group, and let A be a G-algebra, i.e. G acts on A via algebra automorphisms. For a subgroup H of G, A^{H} denotes the algebra of fixed points of H on A. A pointed group on A is a pair (H, α) consisting of a subgroup H of G and a point α of A^{H}. One writes H_{α} instead of (H, α). Let H_{α} be a pointed group on A. Suppressing the subgroup H, 𝔪_{α} denotes the corresponding maximal ideal of A^{H}.

The group G acts on the set of pointed groups on A, where, for g ∈ G and a pointed group H_{α} on A, ^{g}(H_{α}) denotes the pointed group (^{g}H)^{g}_{α} on A; at this ^{g}H denotes the conjugate subgroup gHg^{−1}, and ^{g}α := {^{g}e : e ∈ α}, which is a point of A^{^{g}H}. We refer to this action as conjugation. Let H_{α} be a pointed group on A. The stabilizer of H_{α} in G is denoted by N_{G}(H_{α}). Plainly, H is a normal subgroup of N_{G}(H_{α}). We set \overline{N}_{G}(H_{α}) := N_{G}(H_{α})/H. Moreover, for any subgroup H of G, we set \overline{N}_{G}(H) = N_{G}(H)/H.

(1.3) Let H_{α} and K_{β} be pointed groups on A. Then K_{β} is called a pointed subgroup of H_{α}, written K_{β} ≤ H_{α}, provided that K is a subgroup of H and, for some e ∈ α and some f ∈ β, f appears in an orthogonal decomposition of e into pairwise orthogonal primitive idempotents in A^{K}, i.e. ef = f = fe. The latter condition can also be described in terms of the associated maximal ideals, as follows: Namely, given that K is a subgroup of H, the latter condition holds if and only if

A^{H} ∩ 𝔪_{β} ⊆ 𝔪_{α}.

The relation ≤ on the set of pointed group on A is easily checked to be a partial order relation, which is plainly G-invariant in the sense that K_{β} ≤ H_{α} if and only if ^{g}(K_{β}) ≤ g(H_{α}) for any g ∈ G.

Assume that K is a subgroup of H, and let e ∈ α. Then K_{β} ≤ H_{α} if and only if e ∉ 𝔪_{β}; for if e ∉ 𝔪_{β} then A^{H} ∩ 𝔪_{β} ⊆ 𝔪_{α} since otherwise e ∈ A^{H} = (A^{H} ∩ 𝔪_{β}) + 𝔪_{α} as 𝔪_{α} is a maximal ideal in A^{H}, and so Rosenberg’s lemma ([15] proposition (4.9)) yields a contradiction. The converse implication is immediate as e ∉ 𝔪_{α}.

Now assume that K_{β} ≤ H_{α}, and that L is a subgroup of G such that K ⊆ L ⊆ H. Then there exists some point λ of A^{L} such that

K_{β} ≤ L_{λ} ≤ H_{α}.

In fact, if e ∈ α, and e = i_{1} + i_{2} + … + i_{n} with pairwise orthogonal primitive idempotents i_{1}, i_{2}, …, i_{n} (n ∈ ℕ) in A^{L} then, for some t = 1, …, n, i_{t} ∉ 𝔪_{β} as e ∉ 𝔪_{β}, and we can take λ to be the unique point of A^{L} containing i_{t}. Note that, in this case, K_{β} ≤ L_{λ} by the above observation, and L_{λ} ≤ H_{α} as ei_{t} = i_{t} = i_{t}e.

(1.4) There is another relevant partial order relation on the set of pointed groups on A, which is defined in terms of relative trace maps. Let H and K be subgroups of G such that K is contained in H. Recall that the relative trace map Tr_{K}^{H} : A^{K} → A^{H} is the O-linear map which is given by the assignment

a ↦ ∑_{hK∈H/K} ^{h}a,

and which is independent of the choice of the transversal for the set H/K of cosets hK (h ∈ H). The image of Tr_{K}^{H} is denoted by A_{K}^{H}, which is an ideal in A^{H}, by the Frobenius property of relative trace maps.

Let H_{α} be a pointed group on A, and let K be a subgroup of H. Then H_{α} is called relatively K-projective provided that α ⊆ A_{K}^{H}. Also let β be a point of A^{K}. Then H_{α} is called relatively K_{β}-projective, written H_{β} pr K_{β}, provided that

α ⊆ Tr_{K}^{H}(A^{K}βA^{K}),

where A^{K}βA^{K} denotes the ideal of A^{K} generated by β, which coincides with A^{K}fA^{K} for any f ∈ β. Then H_{α} is relatively K_{β}-projective if and only if for some (any) e ∈ α there exist f ∈ β, a ∈ eA^{K}f, and b ∈ fA^{K}e such that e = Tr_{K}^{H}(ab). (This can be easily shown using the fact that e ∈ α is a primitive idempotent in A^{H}, i.e. eA^{H}e is a local algebra.)

Also the relation pr is a partial order relation on the set of pointed groups on A, which is clearly G-invariant in the above sense, as well.

(1.5) Finally, recall the definition of the Brauer homomorphism. Let H be a subgroup of G. Denote by \overline{A}(H) the residue algebra

A^{H}/J(O)A^{H} + ∑_{K} A_{K}^{H},

where K ranges over all proper subgroups of H. The canonical epimorphism

Br_{H} : A^{H} → \overline{A}(H)

is called the Brauer homomorphism of H (on A). It is clear that \overline{A}(H) = 0 (i.e. Br_{H} = 0) unless H is a p-group as 𝔽 = O/J(O) has characteristic p.

A pointed group P_{γ} on A is called local provided that 𝔪_{γ} contains the kernel of the Brauer homomorphism of P. Thus, if P_{γ} is a local pointed group on A, P is necessarily a p-group. A pointed group P_{γ} on A is local if and only if P_{γ} is not relatively projective to a proper subgroup of P.

(1.6) Let H_{α} be a pointed group on A. A local pointed subgroup P_{γ} of H_{α} is called a defect pointed subgroup of H_{α} (or a defect of H_{α}, for short) provided that H_{α} is relatively P_{γ}-projective; i.e. a pointed group P_{γ} on A is a defect of H_{α} if and only if

P_{γ} is local, P_{γ} ≤ H_{α} and H_{α}pr P_{γ}.

The fundamental result of the theory asserts that any pointed group H_{α} on A has a defect, and any two defects of H_{α} are conjugate in H. Moreover, the defects of H_{α} are precisely the maximal (with respect to the relation ≤) local pointed subgroups of H_{α}. So, as local pointed subgroups are analogues of p-subgroups, defects are analogues of Sylow p-subgroups.

Now we are in a position to formulate the Puig correspondence [12], [13]. A detailed proof of this result can be found in [15].

(1.7) Theorem (Puig correspondence). Let G be a finite group, let A be a G-algebra, let P_{γ} be a local pointed group on A, and let H be a subgroup of G containing P. Set \overline{N} := \overline{N}_{H}(P_{γ}), denote by S the residue algebra A^{P} /𝔪_{γ}, and let π_{γ} : A^{P} → S denote the canonical epimorphism. Then there is a bijection

{α ∈ P(A^{H}) : H_{α} has defect P_{γ}}
→ \{\overline{β} ∈ P(S\overline{N}) : \overline{β} is relatively 1-projective\}
α ↦ π_{γ}(α).

Moreover, if points α ∈ P(A^{H}) and \overline{β} ∈ P(S\overline{N}) correspond under this bijection then

𝔪_{α} = A^{H} ∩ π_{γ}^{−1}(𝔪\overline{β}),

and the composition of the canonical inclusion A^{H} ↪ A^{P} followed by π_{γ} induces an isomorphism of abstract algebras A^{H}/𝔪_{α} → S\overline{N}/𝔪\overline{β}.

We adopt the notation of theorem (1.7). In case points α ∈ P(A^{H}) and \overline{β} ∈ P(S\overline{N}) correspond under the bijection of theorem (1.7), \overline{β} is called the Puig correspondent of α, and, conversely, α is called the Puig correspondent of \overline{β}. We also say that H_{α} is the Puig correspondent of \overline{N}\overline{β}, and vice versa.

(1.8) Assume we have the following inclusion of pointed groups on A

P_{γ} ≤ H_{β} ≤ G_{α},

where P_{γ} is a local, and N_{G}(P_{γ}) ⊆ H. Then P_{γ} is a defect of H_{β} if and only if P_{γ} is a defect of G_{α}. This is the Burry-Carlson-Puig theorem (cf. [15] theorem (20.4)).

Assume that P_{γ} is a defect of H_{β} or of G_{α}, and hence both. Then α determines β uniquely by the above inclusion, and vice versa. G_{α} is then called the Green correspondent of H_{β} (with respect to P_{γ}), and, conversely, H_{β} is called the Green correspondent of G_{α} (with respect to P_{γ}).

Of course, this correspondence is closely related to the Green correspondence for modules as mentionend in the introduction.

Apart from that, we have \overline{N}_{G}(P_{γ}) = \overline{N}_{H}(P_{γ}), by assumption, and it is well-known that G_{α} and H_{β} have the same Puig correspondent with respect to P_{γ}. In fact, this is the way how the Green correspondence is constructed in the proof of [15] theorem (20.1).

(1.9) Recall that an interior G-algebra is a pair (A, Δ) consisting of an algebra A and a homomorphism of groups G → U(A). We usually omit the homomorphism Δ and say that A is an interior G-algebra. We then write g·a, a·g and g·a·h instead of Δ(g)a, aΔ(g) and Δ(g)aΔ(h), respectively, for g, h ∈ G and a ∈ A. This way any interior G-algebra A becomes a G-G-bimodule. Moreover, G acts on A by conjugation, and it is clear that this action turns A into a G-algebra.

Prominent examples of interior G-algebras are blocks of group algebras. Let B be a block of OG, the group algebra of G over O. Then the canonical map G → U(B), g ↦ g1_{B} is a homomorphism of groups turning B into an interior G-algebra. The second class of important examples of interior G-algebras are associated with OG-modules. Let M be an OG-module. Then the algbra E := End_{O}(M) of O-endomorphisms of M is an interior G-algebra, where the corresponding homomorphism of groups G → U(E) maps g ∈ G to the O-automorphism of M which is given by multiplication with g. In particular, E is a G-algebra.

2. Puig Correspondence and Induction

The aim of this section is to demonstrate that an assertion analogous to theorem C also holds for the Puig correspondence. Theorem C will then be an immediate consequence of this result.

2.1. G-Algebras and Induction

Before we state the result, and embark on its proof, let us begin with some further remarks on definition B. Some of these observations will be frequently used in the sequel, partly without further reference.

(2.1) In the subsequent paragraphs (2.1)–(2.4), we keep the following notation: Let G be a finite group, let H_{α} and K_{β} be pointed groups on a G-algebra A, and assume that H_{α} is relatively K_{β}-free. Thus K ⊆ H, there are e ∈ α and f ∈ β such that f(^{h}f) = 0 for all h ∈ H ∖ K, and e = Tr_{K}^{H}(f).

Let also e^{∗} ∈ α. Then e^{∗} = ueu^{−1} for some unit u in A^{H}. Hence f^{∗} := ufu^{−1} ∈ β, f^{∗}( f^{∗}^{h}) = 0 for all h ∈ H ∖ K, and e^{∗} = Tr_{K}^{H}(f^{∗}). (Note, however, that there might as well be some f_{1} ∈ β such that f_{1}( f_{1}^{h}) ≠ 0 for some h ∈ H, and that Tr_{K}^{H}(f_{1}) is not an idempotent at all.) Using this fact, it can be easily cheked that the relation “being relatively free to” on the set of pointed groups on A is transitive. Of course, this relation is also reflexive and antisymmetric, hence a partial order relation. Moreover, this relation is plainly G-invariant. Besides, as already mentioned in the introduction, this relation is a subset of both, the subgroup relation, and the relative projectivitiy relation for pointed groups on A, as well.

(2.2) Now let also H_{α}′ be a pointed group on A, which is relatively free to K_{β}, as well. We claim that α′ = α. In fact, by assumption, there are e′ ∈ α′ and f′ ∈ β such that f′(^{h}f′) = 0 for all h ∈ H ∖ K, and e′ = Tr_{K}^{H}(f′). Thus there is a unit w ∈ A^{K} such that f′ = wfw^{−1}. Hence wf ∈ A^{K}, fw^{−1} ∈ A^{K},

Tr_{K}^{H}(wf) Tr_{K}^{H}(fw^{−1}) = ∑_{xK,yK∈H/K}(^{x}w)(^{x}f)(^{y}f)(^{y}(w^{−1}))
= Tr_{K}^{H}(wfw^{−1}) = Tr_{K}^{H}(f′) = e′,

and similarly Tr_{K}^{H}(fw^{−1}) Tr_{K}^{H}(wf) = Tr_{K}^{H}(w^{−1}f′) Tr_{K}^{H}(f′w) = Tr_{K}^{H}(f) = e. This shows that e and e′ belong to the same point of A^{H}, i.e. β′ = β, as desired.

(2.3) Let I be a subgroup of H, and let i be an idempotent in A^{I} such that i(^{h}i) = 0 for all h ∈ H ∖ I, and e = Tr_{I}^{H}(i). Then i is a primitive idempotent in A^{I} as e is a primitive idempotent in A^{H}. In fact, if i = i_{1} + i_{2} is an orthogonal decomposition of i with idempotents i_{1}, i_{2} ∈ A^{I} then ii_{1} = i_{1} = i_{1}i and ii_{2} = i_{2} = i_{2}i. Hence i_{1}(^{h}i_{1}) = 0 = i_{2}(^{h}i_{2}) for all h ∈ H ∖ I. This plainly implies that e_{1} := Tr_{I}^{H}(i_{1}) and e_{2} := Tr_{I}^{H}(i_{2}) are idempotents in A^{H}. Moreover, e = e_{1} + e_{2} is an orthogonal decomposition. Besides, i_{1} = e_{1}i_{1} and i_{2} = e_{2}i_{2}, so that i_{1} = 0 if and only if e_{1} = 0, and i_{2} = 0 if and only if e_{2} = 0, which proves our claim.

(2.4) Let L be a subgroup of G such that K ⊆ L ⊆ H. As f(^{h}f) = 0 for all h ∈ H ∖K it is clear that l := Tr_{K}^{L}(f) is an idempotent in A^{L}. Besides, l(^{h}l) = 0 for all h ∈ H ∖ L, and Tr_{L}^{H}(l) = Tr_{K}^{H}(f) = e. Thus l is a primitive idempotent in A^{L}, by (2.3). Denote by λ the unique point of A^{L} containing l. Then H_{α} is relatively L_{λ}-free, and L_{λ} is relatively K_{β}-free.

It should be noted that if H_{α} and K_{β} are pointed groups on a G-algebra A such that H_{α} is relatively K_{β}-free then, on the one hand, H_{α} is uniquely determined by K_{β} in the sense of (2.2). On the other hand, however, it is not true in general that, for e ∈ α and f ∈ β, the H-algebra structure of eAe is determined by the K-algebra structure of fAf. Even worse, not even the structure of eAe as abstract algebra is determined by fAf, in general. However, if A is an interior G-algebra then this is the case, as the following lemma demonstrates.

Let H be a subgroup of a finite group G, and let A be an interior H-algebra. Recall that there is a notion of an induced interior G-algebra Ind_{H}^{G} A due to L. Puig (cf. [10] or [15] for details). As abstract algebra Ind_{H}^{G} A is isomorphic to Mat(|G : H|, A). In particular, A and Ind_{H}^{G} A are Morita equivalent as abstract algebras. This notion of induction is compatible with induction of modules in the sense that there is a canonical isomorphism

Ind_{H}^{G} End_{O}(M) →^{∼} End_{O}(Ind_{H}^{G} M)

for any OH-module M.

(2.5) Lemma ([15] Proposition (16.6)). Let H be a subgroup of a finite group G, and let A be an interior G-algebra. Assume that there is an idempotent f ∈ A^{H} such that 1_{A} = Tr_{H}^{G}(f) and f(^{g}f) = 0 for all g ∈ G ∖ H. Then there is an isomorphism of interior G-algebras

A → Ind_{H}^{G}(fAf).

Note that, in a suitable sense, the converse of lemma (2.5) is also valid. Namely, if H is a subgroup of a finite group G and A is an interior H-algebra then there is an idempotent f ∈ (Ind_{H}^{G} A)^{H} such that 1_{Ind_{H}^{G} A} = Tr_{H}^{G}(f), f(^{g}f) = 0 for all g ∈ G ∖ H, and A ≅ f(Ind_{H}^{G} A)f as interior H-algebras.

Finally note that, also for arbitrary G-algebras, definition B is correlated with an induction procedure, namely with induction of modules of skew group algebras (cf. for instance [5]).

2.2. The Result

Our proof of theorem C will be based on the following result:

(2.6) Theorem. Let G be a finite group, and assume we have the following inclusion of pointed groups on a G-algebra A

P_{γ} ≤ H_{β} ≤ G_{α},

where P_{γ} is a defect of G_{α}. Set \overline{N} := \overline{N}_{G}(P_{γ}), \overline{M} := \overline{N}_{H}(P_{γ}), and denote by \overline{M}\overline{β} and \overline{N}\overline{α} the Puig correspondents of H_{β} and G_{α} (with respect to P_{γ}), respectively. Then the following assertions hold:

If G_{α} is relatively H_{β}-free then \overline{N}\overline{α} is relatively \overline{M}\overline{β}-free. If moreover there is a normal subgroup U of G containing P such that γ_{∞}(U) ⊆ H then the converse holds, as well.

We adopt the notation of theorem (2.6). Denote by S the \overline{N}-algebra A^{P} /𝔪_{γ}. Then \overline{N}\overline{α} and \overline{M}\overline{β} both are pointed groups on S. So the assertion “\overline{N}\overline{α} is relatively \overline{M}\overline{β}-free” really makes sense. (Note also that P_{γ} is a maximal local pointed subgroup (i.e. a defect) of H_{β} as P_{γ} is a defect of G_{α}.)

2.3. G-Algebras and Clifford Theory via Thévenaz’s Lifting Theorem

The first implication of theorem (2.6) is fairly trivial. For the proof of the converse implication, we will show that it suffices to assume that U ⊆ H, using straight forward inductive arguments. In this case we reduce further to the situation where precisely one point ρ of A^{U} satisfies U_{ρ} ≤ G_{α}. This will be important in the proof of theorem (2.6) for two reasons. On the one hand, we then have G = U N_{G}(P_{γ}), by the Frattini argument. On the other hand, for any e ∈ α, B := eA^{U}e has a unique point ρ, in this case, and therefore 𝔪_{ρ} = J(B). This reduction is provided by the following Clifford theory type of application of Thévenaz’s lifting theorem.

(2.7) Proposition. Let U be a normal subgroup of a finite group G, and assume we have the following inclusion of pointed groups on a G-algebra A

P_{γ} ≤ U_{ρ} ≤ G_{α},

where P_{γ} is a defect of G_{α}. Set I := N_{G}(U_{ρ}). Then there is a uniquely determined pointed group I_{δ} on A such that

U_{ρ} ≤ I_{δ} ≤ G_{α}.

Moreover, the following assertions hold:

(a) G acts transitively on the set of points τ of A^{U} such that U_{τ} ≤ G_{α},

(b) ρ is the unique point of A^{U} satisfying U_{ρ} ≤ I_{δ}, and I = N_{I}(P_{γ})U,

(c) G_{α} is relatively I_{δ}-free.

Proof. As U_{ρ} ≤ G_{α}, there are e ∈ α and r ∈ ρ such that er = r = re.

Consider the G/U-algebra B := eA^{U}e. Let V/U be a subgroup of G/U. As 1_{B} = e ∈ A_{P}^{G} ⊆ A_{U}^{G} = (A^{U})_{1}^{G/U} ⊆ (A^{U})_{1}^{V/U}, by assumption and the Mackey formula for relative trace maps, and so 1_{B} ∈ B_{1}^{V/U},

B_{1}^{V/U} + J(B)/J(B) = (B/J(B))_{1}^{V/U}

is an ideal in (B/J(B))^{V/U} containing 1_{B/J(B)}. Therefore,

B^{V/U}/B^{V/U} ∩ J(B) ≅ B^{V/U} + J(B)/J(B) = (B/J(B))^{V/U}. (1)

In particular, (B/J(B))^{G/U} is a local algebra as e = 1_{B} is a primitive idempotent in eA^{G}e = B^{G/U}.

For b ∈ B, denote by b̄ the image of b under the canonical epimorphism B → B/J(B). As r̄ is a primitive idempotent in B/J(B), there is a primitive central idempotent ī in B/J(B) such that r̄ī ≠ 0, and hence r̄ī = r̄ as r̄ī is a nonzero idempotent in r̄(B/J(B))r̄.

The action of G/U on B/J(B) permutes the primitive central idempotents of B/J(B), and any orbit yields a nonzero idempotent of (B/J(B))^{G/U}. Consequently, G/U acts transitively on the set of primitive central idempotents of B/J(B). Denote by I/U the stabilizer of ī in G/U. Then ī ∈ (B/J(B))^{I/U}, 1_{B/J(B)} = Tr_{I/U}^{G/U}(ī), and ī(^{gU}ī) = 0 for all gU ∈ (G/U) ∖ (I/U).

Next we show that we have in fact I = N_{G}(U_{ρ}), so that our notation is consistent. At first, let x ∈ I. Thus

(^{xU} r̄)ī = ^{xU}(r̄ī) = ^{xU} r̄.

Hence r̄ and ^{xU} r̄ both are primitive idempotents in (B/J(B))ī. However, (B/J(B))ī is a simple algebra, which thus has a unique point, as ī is a primitive central idempotent in B/J(B). Therefore, r̄ and ^{xU} r̄ belong to the same point of B/J(B), and so r and ^{xU}r belong also to the same point of B = eA^{U}e. But then r and ^{x}r also belong to the same point of A^{U}, i.e. ρ = ρ^{x}. This shows that I ⊆ N_{G}(U_{ρ}).

Conversely, let y ∈ N_{G}(U_{ρ}). Thus ^{y}r ∈ ρ. Hence r and ^{yU}r belong to the same point of B. So there is a unit u in B such that uru^{−1} = ^{yU}r. Therefore,

(^{yU} r̄)ī = ūr̄ū^{−1}ī = ūr̄īū^{−1} = ūr̄ū^{−1} = ^{yU} r̄,

thus

(^{yU} r̄)ī(^{yU}ī) = (^{yU} r̄)(^{yU}ī) = ^{yU}(r̄ī) = ^{yU} r̄,

and in particular ī(^{yU}ī) ≠ 0. This implies that y ∈ I, and hence I = N_{G}(U_{ρ}).

By (1), ī can be lifted to an idempotent i ∈ B^{I/U} = eA^{I}e. Moreover, Tr_{I}^{G}(i) = Tr_{I/N}^{G/N}(i) ≡ e = 1_{B} (mod J(B)), and i(^{g}i) = i(^{gU}i) ∈ J(B) for all g ∈ G ∖ I. Therefore, by Thévenaz’s lifting theorem (in the form of [6] theorem 2.1), u := Tr_{H}^{G}(i) is a unit in B^{G/U} = eA^{G}e, and i^{∗} := u^{−1}i is an idempotent in B^{I/U} = eA^{I}e such that

e = 1_{B} = Tr_{I}^{G}(i^{∗}),

and i^{∗}(^{g}i^{∗}) = 0 for all g ∈ G ∖ I. As e is a primitive idmpotent in A^{G}, this implies that i^{∗} is a primitive idempotent in A^{I}, by (2.3). Thus i is a primitive idempotent in A^{I}, as well, and i and i^{∗} belong to the same point δ of A^{I}, as

i − i^{∗} ∈ J(B) = eJ(A^{U})e ⊆ J(A^{U}) ⊆ J(A^{I}).

(Note that u ≡ 1_{B} (mod J(B)) and U ⊴ I.) Hence G_{α} is relatively I_{δ}-free; in particular, I_{δ} ≤ G_{α}.

Let also I_{δ_{∗}} be a pointed group on A such that U_{ρ} ≤ I_{δ_{∗}} ≤ G_{α}. Thus there are r_{∗} ∈ ρ and i_{∗} ∈ δ_{∗} such that ei_{∗} = i_{∗} = i_{∗}e and r_{∗}i_{∗} = r_{∗} = i_{∗}r_{∗}. Hence i_{∗} ∈ eA^{I}e = B^{I/U}, and r_{∗} ∈ eA^{U}e = B. Therefore, r and r_{∗} also belong to the same point of B. Thus there is a unit u in B such that r_{∗} = uru^{−1}. Hence

r̄_{∗}ī = ūr̄ū^{−1}ī = ūr̄īū^{−1} = ūr̄ū^{−1} = r̄_{∗},

thus r̄_{∗}īī_{∗} = r̄_{∗}ī_{∗} = r̄_{∗}, and in particular īī_{∗} ≠ 0. Thus, as ī is central, īī_{∗} is a nonzero idempotent, both, in ī_{∗}(B/J(B))^{I/U}ī_{∗} and in ī(B/J(B))^{I/U}ī, as well. Therefore, ī = īī_{∗} = ī_{∗} as any of these algebras is local, by (1). Thus i and i_{∗} belong to the same point of B^{I/U} = eA^{I}e. But then i and i_{∗} also belong to the same point of A^{I}, i.e. δ = δ_{∗}, as desired.

Next we prove assertion (b). At first, we show that I = N_{I}(P_{γ})U using the Frattini argument. Since

P_{γ} ≤ U_{ρ} ≤ G_{α},

P_{γ} is a maximal local pointed subgroup (i.e. a defect) of U_{ρ} as P_{γ} is a defect of G_{α}, by assumption. Let x ∈ I. Thus ^{x}(P_{γ}) is a defect of ^{x}(U_{ρ}). However, ^{x}(U_{ρ}) = U_{ρ} as x ∈ I = N_{G}(U_{ρ}), by assumption. Thus there is z ∈ U such that P_{γ} = ^{zx}(P_{γ}), i.e. zx ∈ N_{I}(P_{γ}), and so x ∈ N_{I}(P_{γ})U.

Let also ρ^{∗} be a point of A^{U} such that U_{ρ}^{∗} ≤ I_{δ}. Thus there is r^{∗} ∈ ρ^{∗} such that ir^{∗} = r^{∗} = r^{∗}i. Hence r^{∗} is also a primitive idempotent in iBi. However, iBi is a primary algebra, i.e. iBi has a unique point, as ī(B/J(B))ī = (B/J(B))ī is a simple algebra. Thus r and r^{∗} belong to the same point of iBi = iA^{U}i. But then r and r^{∗} also belong to the same point of A^{U}, i.e. ρ = ρ^{∗}.

It remains to prove assertion (a). Let τ be a point of A^{U} such that U_{τ} ≤ G_{α}. Thus there is t ∈ τ such that t ∈ eA^{U}e = B. We have

0 ≠ t̄= t̄1_{B} = t̄ Tr_{I}^{G}(ī) = ∑_{gI∈G/I} t̄(^{gU}ī).

Thus (^{gU} t̄)i ≠ 0 for some g ∈ G. As above one shows that this implies that ^{g}t and r belong to the same point of A^{U}, i.e. ^{g}τ = ρ, which proves (a). □

An immediate consequence of proposition (2.7) is the following fact, which we state for convenience.

(2.8) Corollary. Let U be a normal subgroup of a finite group G, and assume we have the following inclusion of pointed groups on a G-algebra A

P_{γ} ≤ U_{ρ} ≤ H_{β} ≤ G_{α},

where P_{γ} is a defect of G_{α}. Set I = N_{G}(U_{ρ}), K := N_{H}(U_{ρ}) = H ∩ I, and denote by I_{δ} and K_{σ} the uniquely determined pointed groups on A fulfilling the following inclusions

U_{ρ} ≤ I_{δ} ≤ G_{α},
U_{ρ} ≤ K_{σ} ≤ H_{β}.

Then K_{σ} ≤ I_{δ}. Moreover, G_{α} is relatively H_{β}-free if and only if I_{δ} is relatively K_{σ}-free.

Proof. Note first that P_{γ} is in particular a maximal local pointed subgroup (i.e. a defect) of H_{β} (and of U_{ρ}) as it is a defect of G_{α}, by assumption.

By (1.3), there exists a point δ^{∗} of A^{I} such that

K_{σ} ≤ I_{δ}^{∗} ≤ G_{α}.

In particular, U_{ρ} ≤ I_{δ}^{∗} ≤ G_{α}, and so δ = δ^{∗}, by proposition (2.7). Therefore, K_{σ} ≤ I_{δ}.

It remains to prove the second assertion. Assume first that G_{α} is relatively H_{β}-free. Thus G_{α} is relatively K_{σ}-free, by (2.1), as H_{β} is relatively K_{σ}-free, by proposition (2.7)

(c) (applied to the restricted H-algebra Res_{H}^{G} A). Thus, by (2.4), there is a point δ^{∗} of A^{I} such that G_{α} is relatively I_{δ}^{∗}-free, and I_{δ}^{∗} is relatively K_{σ}-free. In particular,

U_{ρ} ≤ I_{δ}^{∗} ≤ G_{α}.

Hence δ^{∗} = δ, by proposition (2.7), and so I_{δ} is relatively K_{σ}-free.

Conversely, assume that I_{δ} is relatively K_{σ}-free. Then G_{α} is relatively K_{σ}-free, by (2.1), as G_{α} is relatively I_{δ}-free, by proposition (2.7) (c). By (2.4), there is a point β^{∗} of A^{H} such that G_{α} is relatively H_{β}^{∗}-free, and H_{β}^{∗} is relatively K_{σ}-free. However, as also H_{β} is relatively K_{σ}-free, by proposition (2.7) (c), we must have β^{∗} = β, by (2.2). Hence G_{α} is relatively H_{β}-free, completing the proof. □

2.4. A Proof of Theorem (2.6)

Now we are in a position to prove theorem (2.6).

Proof of Theorem (2.6). Denote by S the residue algebra A^{P} /𝔪_{γ}, let π_{γ} : A^{P} → S be the canonical epimorphism, and denote by \overline{γ} := π_{γ}(γ) the unique point of S. Moreover, as the notation of theorem (2.6) suggests, set N = N_{G}(P_{γ}), and set M := N_{H}(P_{γ}).

(⇒) Assume first that G_{α} is relatively H_{β}-free. Thus there are e ∈ α and f ∈ β such that e = Tr_{H}^{G}(f), and f(^{g}f) = 0 for all g ∈ G ∖ H. By theorem (1.7), ē := π_{γ}(e) ∈ π_{γ}(α) = \overline{α}, and f̄:= π_{γ}(f) ∈ π_{γ}(β) = \overline{β}. Thus, as π_{γ} is a homomorphism of \overline{N}-algebras, we have

f̄(^{xP}f̄) = π_{γ}(f(^{x}f)) = 0

for all x ∈ N ∖ M. Moreover, f ∈ Tr_{P}^{H}(A^{P} γA^{P}) as H_{β} is relatively P_{γ}-projective. Thus

ē = π_{γ}(e) = π_{γ}(Tr_{H}^{G}(f)) = Tr\overline{N} ; \overline{M}(π_{γ}(f)) = Tr\overline{N} ; \overline{M}(f̄),

by [15] corollary (14.8). Therefore, \overline{N}\overline{α} is relatively \overline{M}\overline{β}-free.

(⇐) Now assume that \overline{N}\overline{α} is relatively \overline{M}\overline{β}-free, and let U be a normal subgroup of G containing P such that γ_{∞}(U) ⊆ H, which exists, by assumption.

We argue by induction on |G| + |G : H|. At first, we show that we may assume that H is a maximal subgroup of G.

We may assume that |G : H| > 1 since otherwise the assertion is trivial. Suppose that there is a proper intermediate subgroup H ⊂ X ⊂ G. Set \overline{Y} := \overline{N}_{X}(P_{γ}). Thus \overline{M} ⊆ \overline{Y} ⊆ \overline{N}. By (2.4), there is a point \overline{ω} of S\overline{Y} such that \overline{N}\overline{α} is relatively \overline{Y} \overline{ω}-free, and \overline{Y} \overline{ω} is relatively \overline{M}\overline{β}-free. In particular,

\overline{M}\overline{β} ≤ \overline{Y} \overline{ω} ≤ \overline{N}\overline{α}.

Moreover, \overline{Y} \overline{ω} is relatively 1-projective as \overline{M}\overline{β} is relatively 1-projective, by theorem (1.7). Denote by X_{ω} the Puig correspondent of \overline{Y} \overline{ω}. Thus ω is the unique point of A^{X} such that X_{ω} has defect P_{γ} and π_{γ}(ω) = \overline{ω}.

We claim that H_{β} ≤ X_{ω} ≤ G_{α}. Let a ∈ A^{G} ∩ 𝔪_{ω}. Thus π_{γ}(a) ∈ S\overline{N} ∩ 𝔪\overline{ω} ⊆ 𝔪\overline{α}, by theorem (1.7). Hence a ∈ A^{G} ∩ π_{γ}^{−1}(𝔪\overline{α}) = 𝔪_{α}, again by theorem (1.7). Thus A^{G} ∩ 𝔪_{ω} ⊆ 𝔪_{α}, and so X_{ω} ≤ G_{α}. Similarly, one shows that \overline{M}\overline{β} ≤ \overline{Y} \overline{ω} implies that H_{β} ≤ X_{ω}.

By assumption, we have |X| + |X : H| < |G| + |G : H|. Moreover, X ∩ U is a normal subgroup of X containing P such that γ_{∞}(X ∩ U) ⊆ γ_{∞}(U) ⊆ H. Therefore, X_{ω} is relatively H_{β}-free, by induction.

On the other hand, |G| + |G : X| < |G| + |G : H|. Besides, γ_{∞}(U) ⊆ H ⊆ X. Thus G_{α} is relatively X_{ω}-free, again by induction. Hence G_{α} is relatively H_{β}-free, by (2.1).

Therefore, we may assume that H is a maximal subgroup of G. We claim that H∩U ⊴G. Suppose not. Then H ∩ U is a proper subgroup of U as U ⊴ G. Moreover, as H ⊆

N_{G}(H ∩ U) and H is a maximal subgroup of G, we then must have N_{G}(H ∩ U) = H. By assumption, γ_{∞}(U) ⊆ H ∩ U. Thus H ∩ U/γ_{∞}(U) is a proper subgroup of the nilpotent group U/γ_{∞}(U). Therefore, H ∩ U/γ_{∞}(U) is a proper subgroup of

N_{U/γ_{∞}(U)}((H ∩ U)/γ_{∞}(U)) = N_{U}(H ∩ U)/γ_{∞}(U).

However, N_{U}(H ∩ U) = N_{G}(H ∩ U) ∩ U = H ∩ U, which is a contradiction. Therefore, H ∩ U ⊴ G. Thus we may replace U by H ∩ U and assume that U ⊆ H. Hence there is a point ρ of A^{U} such that

P_{γ} ≤ U_{ρ} ≤ H_{β},

by (1.3). Set I = N_{G}(U_{ρ}). Then proposition (2.7) yields uniquely determined pointed groups I_{δ} and (H ∩ I)_{σ} on A such that

U_{ρ} ≤ I_{δ} ≤ G_{α},
U_{ρ} ≤ (H ∩ I)_{σ} ≤ H_{β}.

Thus P_{γ} ≤ U_{ρ} ≤ (H ∩ I)_{σ} ≤ I_{δ} ≤ G_{α}, by corollary (2.8), and so P_{γ} is a maximal local pointed subgroup (i.e. a defect) of U_{ρ}, (H ∩ I)_{σ} and I_{δ}, as well, as P_{γ} is a defect of G_{α}, by assumption.

Set \overline{L} := \overline{N}_{I}(P_{γ}), and set \overline{V} := \overline{N}_{U}(P_{γ}) ⊴ \overline{N}. Then \overline{N}_{H∩I}(P_{γ}) = \overline{N}_{H}(P_{γ}) ∩ \overline{N}_{I}(P_{γ}) = \overline{M} ∩ \overline{L}. Denote by \overline{V} \overline{ρ}, (\overline{M} ∩ \overline{L})\overline{σ} and \overline{L}\overline{δ} the Puig correspondents of U_{ρ}, (H ∩ I)_{σ} and I_{δ} (with respect to P_{γ}), respectively. Then we clearly have the following inclusions of pointed groups on S

\overline{V} \overline{ρ} ≤ \overline{L}\overline{δ} ≤ \overline{N}\overline{α},
\overline{V} \overline{ρ} ≤ (\overline{M} ∩ \overline{L})\overline{σ} ≤ \overline{M}\overline{β}.

Moreover, 1\overline{γ} ≤ \overline{M}\overline{β} ≤ \overline{N}\overline{α}, and \overline{M}\overline{β}, \overline{N}\overline{α} are relatively 1-projective, by theorem (1.7); hence both pointed groups have defect 1\overline{γ}.

Let x ∈ N_{I}(P_{γ}). Then ρ = ρ^{x} as I = N_{G}(U_{ρ}), and so

\overline{ρ} = π_{γ}(ρ) = π_{γ}( ρ^{x}) = ^{xP} \overline{ρ}.

Thus \overline{L} = \overline{N}_{I}(P_{γ}) ⊆ N\overline{N}(\overline{V} \overline{ρ}). Conversely, asume that xP ∈ \overline{N} fixes \overline{ρ}. Thus π_{γ}(ρ) = \overline{ρ} = ^{xP} \overline{ρ} = π_{γ}( ρ^{x}). Besides, ^{x}(U_{ρ}) = U ρ^{x} has defect ^{x}(P_{γ}) = P_{γ} as P_{γ} is a defect of U_{ρ} and x ∈ N = N_{G}(P_{γ}). Therefore, ρ = ρ^{x}, i.e. x ∈ N_{G}(U_{ρ}) = I, by theorem (1.7), and so xP ∈ \overline{L}. This shows that \overline{L} = N\overline{N}(\overline{V} \overline{ρ}).

Hence \overline{L}\overline{δ} is relatively (\overline{M} ∩\overline{L})\overline{σ}-free, by corollary (2.8) (d), as \overline{N}\overline{α} is relatively \overline{M}\overline{β}-free, by assumption.

Suppose that |I| < |G|. Hence |I| + |I : H ∩ I| < |G| + |G : H| as I/H ∩ I → G/H, x(H ∩I) → xH is an injective map. Moreover, I ∩U is a normal subgroup of I containing P such that γ_{∞}(I ∩ U) ⊆ γ_{∞}(U) ∩ I ⊆ H ∩ I. Thus I_{δ} is relatively (H ∩ I)_{σ}-free, by induction. But then G_{α} is relatively H_{β}-free, by corollary (2.8), and we are done in this case.

Therefore, we may assume that G = I, and so I_{δ} = G_{α} and H_{β} = (H ∩ I)_{σ}. Hence G = N_{G}(P_{γ})U = NU, and ρ is the unique point of A^{U} satisfying U_{ρ} ≤ G_{α}, by proposition (2.7) (b).

By assumption, there are ē ∈ \overline{α} and f̄ ∈ \overline{β} such that ē = Tr_{H}^{G}(f̄), and f̄(^{xP}f̄) = 0 for all xP ∈ \overline{N} ∖ \overline{M}. Moreover, by thereom (1.7), there is e ∈ α such that π_{γ}(e) = ē.

As \overline{N}\overline{α} is relatively 1-projective, by theorem (1.7), we have ē ∈ S\overline{N} ; 1 ⊆ S\overline{M} ; 1. Thus ēS\overline{M} ē = ēS\overline{M} ; 1 ē. As S = A^{P} γA^{P} + 𝔪_{γ}/𝔪_{γ}, [15] proposition (14.7) yields that π_{γ} maps A_{P}^{H} onto S\overline{M} ; 1. Therefore, π_{γ} maps eA_{P}^{H}e onto ēS\overline{M} ; 1 ē = ēS\overline{M} ē. In particular, π_{γ} maps eA^{H}e onto ēS^{M} ē, and moreover

eA^{H}e = eA_{P}^{H}e + (eA^{H}e ∩ ker π_{γ}).

Thus there is a primitive idempotent f ∈ eA^{H}e such that π_{γ}(f) = f̄. Then f is also a primitive idempotent in A^{H}, which is thus contained in a unique point β′ of A^{H}. By Rosenberg’s lemma, we must have f ∈ A_{P}^{H} as π_{γ}(f) = f̄ ≠ 0. (Note that eA_{P}^{H}e is an ideal in eA^{H}e independent of whether or not e ∈ A_{P}^{H}. Moreover, eA_{P}^{H}e ⊆ A_{P}^{H}.) Thus H_{β}′ is relatively P -projective. On the other hand, A^{H} ∩ 𝔪_{γ} ⊆ 𝔪_{β}′, i.e. P_{γ} ≤ H_{β}′, by (1.3), as f ∉ ker π_{γ} = 𝔪_{γ}. Therefore, P_{γ} is a defect of H_{β}′. But then theorem (1.7) implies that f ∈ β′ = β, as f̄ = π_{γ}(f) ∈ π_{γ}(β′), and so π_{γ}(β) = π_{γ}(β′).

Consider the G-algebra eA^{U}e. Any point ε of B is contained in a unique point ε^{∗} of A^{U}. Let j ∈ ε ⊆ ε^{∗}, where ε is some point of B. Then ej = j = je as j ∈ B = eA^{U}e. Thus U_{ε}^{∗} ≤ G_{α}. As ρ is the unique point of A^{U} satisfying U_{ρ} ≤ G_{α}, we obtain ε^{∗} = ρ. Therefore, ε is the unique point of B. Hence B has a unique maximal ideal J(B) = 𝔪_{ε}. Moreover,

B ∩ 𝔪_{ρ} = eA^{U}e ∩ 𝔪_{ε}^{∗} = 𝔪_{ε} = J(B).

As P_{γ} ≤ U_{ρ}, we have A^{U} ∩ 𝔪_{γ} ⊆ 𝔪_{ρ}, and thus B ∩ 𝔪_{γ} ⊆ B ∩ 𝔪_{ρ} = J(B).

Let T be a transversal for N/M. As G = UN, and so in particular G = HN, we have |G : H| = |N : H ∩N| = |N : M|. Therefore, T is a transversal for G/H, as well. Besides, the t̄:= tP (t ∈ T) form a transversal for \overline{N}/\overline{M}.

Let s, t ∈ T, s ≠ t. Then

(^{s}f)(^{t}f) ⊆ eA^{^{s}H∩^{t}H}e ⊆ eA^{U}e = B

as U ⊆ H and U ⊴ G, and so U ⊆ ^{s}H ∩^{t}H. Apart from that,

π_{γ}((^{s}f)(^{t}f)) = (s̄(π_{γ}(f)))(t̄(π_{γ}(f))) = ( f̄^{s̄})( f̄^{t̄}) = 0

as T ⊆ N and s̄^{−1}t̄∉ \overline{M}. Thus (^{s}f)(^{t}f) ∈ B ∩ 𝔪_{γ} ⊆ J(B) for all s, t ∈ T, s ≠ t. Moreover,

π_{γ}(Tr_{H}^{G}(f)) = ∑_{t∈T} π_{γ}(^{t}f) = ∑_{t∈T} f̄^{t̄} = Tr\overline{N} ; \overline{M}(f̄) = ē = π_{γ}(e) = 1_{B}.

Hence 1_{B} − Tr_{H}^{G}(f) ∈ B ∩ 𝔪_{γ} ⊆ J(B). By lemma A, u := Tr_{H}^{G}(f) is a unit in B, and the u^{−1}(^{t}f), t ∈ T, are pairwise orthogonal idempotents in B such that e = 1_{B} = ∑_{t∈T} u^{−1}(^{t}f). As u ∈ B^{G}, and so u^{−1} ∈ B^{G}, f^{∗} := u^{−1}f is even an idempotent in B^{H}. Moreover, for t ∈ T and g ∈ tH, we have

f^{∗}^{g} = f^{∗}^{t} = ^{t}(u^{−1}f) = u^{−1}(^{t}f).

Therefore, e = Tr_{H}^{G}(f^{∗}), and f^{∗}( f^{∗}^{g}) = 0 for all g ∈ G ∖ H.

As u ≡ 1_{B} (mod J(B)), and so f − f^{∗} ∈ B^{H} ∩ J(B) ⊆ J(B^{H}), f^{∗} is a primitve idempotent in B^{H}, and f and f^{∗} belong to the same point of B^{H} = eA^{H}e. But then f and f^{∗} also belong to the same point of A^{H}, i.e. f^{∗} ∈ β. Therefore, G_{α} is relatively H_{β}-free, as we wanted to show. □

3. A Proof of Theorem C and Applications

The aim of this section is to prove the results stated in the introduction. The proof of corollary D, which implies theorem C, is now immediate.

Proof of Corollary D. As N_{G}(P_{γ}) ⊆ K, by assumption, we have \overline{N} := \overline{N}_{G}(P_{γ}) = \overline{N}_{K}(P_{γ}). Similarly, \overline{M} := \overline{N}_{H}(P_{γ}) = \overline{N}_{L}(P_{γ}) as N_{H}(P_{γ}) ⊆ L, by assumption.

Assume either that G_{α} is relatively H_{β}-free or that K_{κ} is relatively L_{λ}-free. In particular, this yields either that H_{β} ≤ G_{α} or that L_{λ} ≤ K_{κ}.

By the remark subsequent to corollary D, P_{γ} is a defect of L_{λ}, K_{κ}, and H_{β}, as well. Moreover, K_{κ} is the Green correspondent of G_{α} (with respect to P_{γ}), and L_{λ} is the Green correspondent of H_{β} (with respect to P_{γ}). So that the Puig correspondent \overline{N}\overline{ν} of K_{κ} (with respect to P_{γ}) is also the Puig correspondent of G_{α}, and the Puig correspondent \overline{M}\overline{μ} of L_{λ} (with respect to P_{γ}) is also the Puig correspondent of H_{β} (cf. [15] theorem (20.1) and its proof for more details).

Thus theorem (2.6) applied either to the G-algebra A or the restricted K-algebra Res_{K}^{G} A yields that \overline{N}\overline{ν} is relatively \overline{M}\overline{μ}-free, in any case. In particular, \overline{M}\overline{μ} ≤ \overline{N}\overline{ν}. As in the proof of theorem (2.6) one shows that this implies that, in any case, we have actually both inclusions, H_{β} ≤ G_{α} and L_{λ} ≤ K_{κ}, as well, using Rosenberg’s lemma.

Hence G_{α} is relatively H_{β}-free, by the converse implication of theorem (2.6), in case the additional hypothesis of assertion (b) is satisfied, and K_{κ} is relatively L_{λ}-free, by the converse implication of theorem (2.6) applied to the restricted K-algebra Res_{K}^{G} A, if the additional hypothesis of assertion (a) holds. □

Next we deduce corollary E.

Proof of Corollary E. Set N := N_{G}(P_{γ}), M := N_{H}(P_{γ}) = H ∩ N, and X := P O^{p}(N) ⊴ N. Thus |N : X| is a power of p, and P ⊆ X ⊆ H ∩ N = M, by assumption. Hence there is a point ω of A^{X} such that

P_{γ} ≤ X_{ω} ≤ H_{β}.

Similarly, there is a point μ of A^{M} such that

X_{ω} ≤ M_{μ} ≤ H_{β}.

Moreover, as H_{β} ≤ G_{α}, by assumption, there is then also a point ν of A^{N} such that

M_{μ} ≤ N_{ν} ≤ G_{α}.

Thus P_{γ} is a maximal local pointed subgroup (i.e. a defect) of both, M_{μ} and N_{ν}, as well, as P_{γ} is a defect of G_{α}, by assumption. In particular, M_{μ} and N_{ν} both are relatively P -projective, and hence relatively X-projective. Therefore, by Green’s theorem (in the form of [15] corollary (23.3)), there is a point ω^{∗} of A^{X} such that N_{ν} is relatively X_{ω}^{∗}-free. In particular, X_{ω}^{∗} ≤ N_{ν}. Thus there is some n ∈ N such that ^{n}ω^{∗} = ω, by proposition (2.7) (a). Therefore, N_{ν} = ^{n}(N_{ν}) is relatively free to

n(X_{ω}^{∗}) = X^{n}_{ω}^{∗} = X_{ω}

as n ∈ N and X ⊴ N.

Similarly, one shows that M_{μ} is relatively X_{ω_{1}}-free for some point ω_{1} of A^{X}. However, as X_{ω} ≤ M_{μ}, this implies as above that ^{m}ω_{1} = ω for some m ∈ M, and thus M_{μ} is relatively X_{ω}-free, as well.

By (2.4), there exists a point μ^{∗} of A^{M} such that M_{μ}^{∗} is relatively X_{ω}-free, and N_{ν} is relatively M_{μ}^{∗}-free. However, we then have M_{μ} = M_{μ}^{∗}, by (2.2), and so N_{ν} is relatively M_{μ}-free.

Moreover, by assumption, P ⊆ U ⊴ G, and γ_{∞}(U) ⊆ H. Thus G_{α} is relatively H_{β}-free, by theorem C (resp. corollary D (b)). □

It remains to prove corollary F.

Proof of Corollary F. By assumption, there exists a point γ of A^{Q} such that Q_{γ} is a defect of G_{α}. Set N := N_{G}(Q), and set M = N_{H}(Q) = H ∩ N. Moreover, denote by N_{ν} the Green correspondent of G_{α} with resect to Q_{γ}, i.e. ν is the unique point of A^{N} satisfying

Q_{γ} ≤ N_{ν} ≤ G_{α}

(cf. [15] theorem (20.1)). Besides, N_{ν} has defect Q_{γ}, as well.

As Q ⊆ H and |G : H| is a power of p, by assumption, |Q| |G : H| divides |P |. Thus, as Q ⊆ P, |G : H| divides |P : Q|. Therefore,

|P H| = \frac{|P | |H|}{|P ∩ H|} ≥ |H| |P : Q| ≥ |H| |G : H| = |G|

as Q ⊆ P ∩ H, by assumption. Hence G = P H. In particular, G = NH as Q ⊴ P, i.e. P ⊆ N, by assumption. Thus |N : M| = |G : H| is a power of p, by assumption, and so M contains some Hall p′-subgroup K of N. (Recall that, by assumption, N is p-solvable.)

Let e ∈ ν. As N_{ν} is relatively Q-projective, we have e ∈ A_{Q}^{N} = (A^{Q})_{1}^{N/Q}. By [6] theorem 3.1, there is an idempotent f ∈ (A^{Q})^{K/Q} = A^{K} such that

e = Tr_{K/Q}^{N/Q}(f) = Tr_{K}^{N}(f),

and f(^{x}f) = f(^{xQ}f) = 0 for all x ∈ N ∖ K. (We leave it to the reader to verifiy that the proof of [6] theorem 3.1 is also valid for G-algebras which are not necessarily torsion-free as O-modules.) By (2.3), f is a primitive idempotent in A^{K}. Denote by κ the unique point of A^{K} containing f. Thus N_{ν} is relatively K_{κ}-free. As K ⊆ M ⊆ N, there is a point μ of A^{M} such that N_{ν} is relatively M_{μ}-free, and M_{μ} is relatively K_{κ}-free. In particular, N_{ν} is relatively M_{μ}-projective. Thus there is x ∈ N such that

Q_{γ} ≤ ^{x}(M_{μ}) ≤ N_{ν},

by [15] lemma (18.2), and so Q_{γ} is a defect of ^{x}(M_{μ}).

Denote by ^{x}(H_{β}) the Green correspondent of ^{x}(M_{μ}) with respect to Q_{γ}. (Note that N_{^{x}H}(Q_{γ}) ⊆ N_{^{x}H}(Q) = ^{x}(N_{H}(Q)) = ^{x}M as x ∈ N = N_{G}(Q).) Thus

Q_{γ} ≤ ^{x}(M_{μ}) ≤ ^{x}(H_{β}).

Moreover, N_{ν} is relatively ^{x}(M_{μ})-free, as well.

As G = NH and x ∈ N, we also have G = N(^{x}H), and so the additional hypothesis of corollary D (b) is satisfied, by an observation subsequent to theorem C. Therefore, G_{α} is relatively ^{x}(H_{β})-free, by corollary D (b). This implies that G_{α} is relatively H_{β}-free, and we are done. □

Acknowledgements

I wish to thank the referee for suggesting to state the main result of this note in the simple form of theorem C, pointing out some redundancies in the proof of theorem (2.6) in an earlier version of this paper, and suggesting to inlcude also the case of algebras which are not necessarily torsion-free over O. Moreover, I am grateful to Burkhard Külshammer for help with the latter suggestion by telling me some references in the literature which are concerned with this more general setup, and for help with some other questions of mine.

18 HUBERT FOTTNER

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