Central units in blocks and the odd Z_{p}^{∗}-theorem Geoffrey R. Robinson, Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland (e-mail: g.r.robinson@abdn.ac.uk)

ABSTRACT: We construct some units of order p in certain p-adic group rings of finite groups, which have relevance to the odd analogue of Glauberman’s Z^{∗}- theorem, but whose construction may have wider applicability and interest.

INTRODUCTION: We are concerned here with the situation that p is an odd prime, G is a finite group, and x is an element of order p which is isolated in a Sylow p-subgroup, P, of G. That is to say, x^{G} ∩ P = {x}, where x^{G} is the conjugacy class of x. The odd analogue of Glauberman’s Z^{∗}-theorem would be to prove that G = O_{p′}(G)H in this situation, where H = C_{G}(x). This is known to be true by the Classification of Finite Simple Groups, but an independent proof of the fact would have considerable theoretical interest (and would have potential applications to revision programmes, being relevant to issues such as the p-analogue of L-balance, and the general structure of groups with Abelian Sylow p-subgroups, for example). It might also be viewed as a test case for the future efficacy of block-theory in further understanding the structure of finite groups.

In the course of our investigations, we obtain some general techniques for constructing units of order p in p-adic group rings of finite groups which may have wider interest.

The following Lemma is well-known, and can be proved using Sylow’s Theorem and the isolation property:

LEMMA 1: i) x is central in any p-subgroup of G containing it, and x commutes with none of its distinct G-conjugates.

ii) If M is a proper subgroup of G which contains x, then any two conjugates of x contained in M are already conjugate within M, and x is isolated in any Sylow p-subgroup of M containing it.

iii) Whenever T is a subgroup of H, and T^{g} ≤ H for some g ∈ G, we may write g = ch for some c ∈ C_{G}(T) and h ∈ H. In particular, if T is a p-subgroup of H containing x, we have N_{G}(T) ≤ H.

iv) Whenever M is a subgroup of G containing x and N ⊲M, then xN is isolated in any Sylow p-subgroup of M/N which contains it and C_{M/N}(xN) = N(M ∩ H)/N.

We begin our first foray into block theory. Let (𝕂, R, k) be a p-modular system such that 𝕂 contains a primitive |G|-th root of unity, and let η be a primitive p-th root of unity in 𝕂. For an irreducible character χ of 𝕂G, we set

e_{χ} = ∑_{g∈G} \frac{χ(1)χ(g^{−1})}{|G|}g,

which is a primitive idempotent of Z(𝕂G). We also let ω_{χ} denote the central character whose value on an element z of Z(𝕂G) is \frac{χ(z)}{χ(1)}. We let K_{x} denote the class sum of x in the group algebra ℤG.

LEMMA 2: i) Brauer correspondence yields a defect-group preserving bijection between blocks of RG whose defect groups contain a conjugate of x and blocks of RH. Furthermore, if b is a block of RH and b^{G} = B, then λ_{B} = Br_{⟨x⟩} ∘ λ_{b}, where λ_{B} : Z(kG) → k is the algebra homomorphism associated to B.

ii) An irreducible character χ of G lies in a block of RG whose defect group contains a conjugate of x if and only if χ(x) ≠ 0.

PROOF: i) If D is a p-subgroup of G containing x, then N_{G}(D) = N_{H}(D), so the first claim follows from Brauer’s First Main Theorem.

ii) follows from part i), together with the facts that x^{G} ∩ H = {x} and (by Brauer’s Second Main Theorem) an irreducible character which lies in a block whose defect group does not contain a conjugate of x vanishes identically on the p-section of x.

¿From now on, we assume that G is a (putative) minimal counterexample to the “Z_{p}^{∗}-theorem” for odd p. It is again well-known (and easy to check) that there are two possibilities for the structure of G- either G is a non-Abelian simple group of order divisible by p or G = G′⟨x⟩, where G′ is a non-Abelian simple group of order divisible by p and x ∉ G′ ( and, if G has Abelian Sylow p-subgroups, the latter case must occur).

Furthermore, whenever M is a proper subgroup of G containing x, we have M = O_{p′}(M)(H ∩ M), by the minimality of G.

We let E denote the sum of the block idempotents of Z(RG) whose defect groups contain (a conjugate of )x (we will obtain an explicit description of E later, but note for the moment that it is a rational combination of group elements). We set A = Eℤ_{p}G, B = A/pA and ∗ denote images in B. For each proper p-local subgroup M containing x, the results of [5] show that Z(C) and Z(GF(p)(M ∩H)) are isomorphic via the Brauer homomorphism Br_{⟨x⟩}, where C is the sum of those blocks of GF(p)M whose defect groups contain x. This easily implies that Br_{⟨x⟩} induces an isomorphism between \frac{Z(B)}{Tr_{1}^{G}(B)} and Z(GF(p)H).

When G ≠ G′, we know from Lemma 2 above that K_{x}^{∗} is a unit of B. Now K_{x}^{∗}Tr_{1}^{G}(B) is in both the span of the p-singular elements ( as it is a linear combination of elements in the coset xG′) and in the span of p-regular elements ( as it is in the ideal Tr_{1}^{G}(GF (p)G)). Hence Tr_{1}^{G}(B) = 0, so that Z(B) ≅ Z(GF(p)H) in that case. Thus we have:

THEOREM 1:i) In the case G ≠ G′, there is a bijection between irreducible characters of H and irreducible characters of G which do not vanish at x. ii) In the case G = G′, the number of irreducible characters of G which do not vanish at x is at least k(H).

Our next aim is to generalize to other blocks results which we obtained in [4] and [6] for principal blocks ( for primes p > 3 or p = 3 and G ≠ G′ ) the corresponding results for p = 3 and G = G′ for principal blocks being established later by M.Hertweck in his 2004 Habilitationsschrift [1]).

LEMMA 3: i) Let M be a proper subgroup of G containing x and let C_{x} denote the class sum of x in M in the group algebra ℤM. Then C_{x}^{p} = C_{x^{i}}^{p} for 1 ≤ i ≤ p − 1.

ii) For 1 ≤ i ≤ p − 1, K_{x^{i}}^{p} − K_{x}^{p} = Tr_{1}^{G}(z) for some z ∈ ℤG which has support on p-singular elements. In particular, K_{x^{i}}^{p} − K_{x}^{p} ∈ pℤG.

PROOF: Choose an integer i with 1 < i < p.

i) We note that if y is p-regular, then (by the standard formula for products of class sums) the coefficient of y in K_{x}^{p} is given by

\frac{|G|^{p−1}}{|H|^{p}} ∑_{χ∈Irr(G)} \frac{χ(x)^{p}χ(y^{−1})}{χ(1)^{p−1}}.

This is clearly algebraically conjugate to

\frac{|G|^{p−1}}{|H|^{p}} ∑_{χ∈Irr(G)} \frac{χ(x^{i})^{p}χ(y^{−1})}{χ(1)^{p−1}},

which is the coefficient of y in K_{x^{i}}^{p}. Since both quantities are rational integers, they are equal. A similar argument within a proper subgroup M containing x shows that C_{x^{i}}^{p} − C_{x}^{p} is a ℤ-combination of p-singular elements of M. On the other hand, M = O_{p′}(M)C_{M}(x) by the minimality of G, so that C_{x}^{p} and C_{x^{i}}^{p} are both ℤ-combinations of p-regular elements of M , since all M-conjugates of x lie in the coset O_{p′}(M)x ( and an analogous statement holds for x^{i}). Thus C_{x}^{p} = C_{x^{i}}^{p}.

ii) By part i), it suffices to prove that for each p-singular element y ∈ G, the number of p-tuples (x_{1}, x_{2}, … x_{p}) of conjugates of x whose product ( in the given order) is y differs from the corresponding number ( with x^{i} replacing x) by an integer multiple of |C_{G}(y)|. By the argument in part i) (and part ii) of Lemma 1), (in either case) any such p-tuple whose product is y must generate G, so C_{G}(y) acts semi-regularly on such p-tuples.

REMARK: Recall that E is the sum of the block idempotents of blocks of RG whose defect groups contain (a conjugate of ) x, then as E ≡ (K_{x^{2}}/K_{x})^{p}(modpℤ_{p}G), we have

E = lim_{n→∞}(K_{x^{2}}/K_{x})^{p^{n}}

- we ask the reader to entertain a slight abuse here: we have EK_{x} = K_{x}, and K_{x} is a unit in Eℤ_{p}G.

We will construct a “primitive p-th root” u_{x} of E in Z(ℤ_{p}G) and examine some consequences of its existence. While it will turn out in the particular case under consideration that the unit u_{x} has a simpler expression than the one initially given, the initial construction may have have independent interest.

Let us remark that a sequence (z_{n}) of elements of Z(𝕂G) converges (in the metric on 𝕂G naturally induced by the J(R)-adic metric on 𝕂) if and only if, for each irreducible character χ of 𝕂G, the sequence (ω_{χ}(z_{n})) is Cauchy (furthermore, if the latter sequence has limit ℓ_{χ} for each χ, then (z_{n}) has limit ∑_{χ∈Irr(G)} ℓ_{χ}e_{χ}).

Most of our calculations take place in ℚ_{p}[η], and we note that J(R)∩ℤ_{p}[η] = (1 − η)ℤ_{p}[η].

Let us also observe that ( as is well-known), when r is a positive integer

and the binomial coefficient ( r^{\frac{1}{p}} ) is expressed as a rational number in lowest

terms, then the denominator is p^{r+ \frac{r−σp(r)}{p−1}} , where σ_{p}(r) is the sum of the digits in the p-adic expansion of r, the second term arising from the exact power of p dividing r!. Since ((1 − η)^{p−1}) = (p), we may extract p-th roots via the well-known:

LEMMA 4: Let γ ∈ (1 − η)^{p+1}ℤ_{p}[η] = (p(1 − η)^{2})ℤ_{p}[η]. Then we have

(∑_{r=0}^{∞} ( r^{\frac{1}{p}} ) γ^{r})^{p} = (1 + γ),

and the leftmost sum lies in ℤ_{p}[η].

The key idea of what immediately follows is that if X is a finite group, and t is an element of Z(ℤ_{p}X), with ω_{μ}(t)^{p} ≡ 1 (mod(1 − η)^{p+1}) for each irreducible character μ of X, then the expected power series expansion for

a = (t^{−p})^{1/p} = (e − (e − t^{−p}))^{1/p} = e + ∑_{r=1}^{∞} ( r^{\frac{1}{p}} ) (t^{−p} − e)^{r}

converges in ℚ_{p}X, with respect to the appropriate p-adic metric, where e is the identity element of the ideal tℚ_{p}X. Also, (at)^{p} = e. However, it seems that additional information is needed to ensure that a ∈ ℤ_{p}X. Fortunately, in the case we are considering, we have sufficient information to ensure the desired integrality.

We let v_{x} = \frac{K_{x^{2}}}{K_{x}} , which is a central unit in Eℤ_{p}G. We note that for each irreducible character χ of G with χ(x) ≠ 0, we have, by part ii) of Lemma 3:

ω_{χ}(v_{x}^{p}) ≡ 1(mod\frac{|G|_{p}}{χ(1)_{p}}).

Now if pχ(1)_{p} = |G|_{p}, then χ lies in a block of defect 1 of RG, which must have defect group ⟨x⟩. Since N_{G}(⟨x⟩) = C_{G}(x), by the theory of blocks with cyclic defect group, such a block contains p irreducible characters, χ(x)^{p} = χ(x^{2})^{p}, ω_{χ}(v_{x}^{p}) = 1 and ω_{χ}(v_{x}) = η_{χ} for some p-th root of unity η_{χ}.

Otherwise, we certainly have ω_{χ}(v_{x}^{p}) ≡ 1 (mod p^{2}). Hence, using Lemma 4 and evaluating the central character ω_{χ} (for each irreducible character χ on the series), we see that

w_{x} = E + ∑_{r=1}^{∞} ( r^{\frac{1}{p}} ) (v_{x}^{−p} − E)^{r}

converges in Z(ℚ_{p}G), and that (w_{x}v_{x})^{p} = E. We set u_{x} = w_{x}v_{x}.

We will prove that u_{x} lies in ℤ_{p}G. It suffices to prove that w_{x} lies in ℤ_{p}G.

LEMMA 5: i) For r ≥ 2, ( r^{\frac{1}{p}} ) (v_{x}^{−p} − E)^{r} lies in ℤ_{p}G. If p ≥ 5 and r > 2

or if p ≥ 3 and r > 3, the given expression lies in pℤ_{p}G. ii) u_{x} ≠ E.

PROOF: i) In 𝕂G, we have

( r^{\frac{1}{p}} ) (v_{x}^{−p} − E)^{r} = ∑_{\{χ∈Irr(G):χ(x)≠0\}} ( r^{\frac{1}{p}} ) (ω_{χ}(v_{x})^{−p} − 1)^{r}e_{χ}.

The coefficient of e_{χ} is 0 if χ lies in a block of defect 1, and otherwise is (in R) a multiple of

\frac{|G|_{p}}{χ(1)_{p}} (\frac{|G|_{p}}{χ(1)_{p}})^{r−1}p^{−r− \frac{r−σp(r)}{p−1}} .

Since r ≥ 2, this coefficient is (in all cases) a multiple of \frac{|G|_{p}}{χ(1)_{p}} . Since \frac{|G|_{p}}{χ(1)_{p}} e_{χ} lies in RG for each such χ, part i) follows. ii) Since v_{x}^{−p} − E lies in pℤ_{p}G by part ii) of Lemma 3, we deduce (also using part i) above) that u_{x} lies in ℤ_{p}G. We note that in the case G ≠ G′, the element w_{x} lies in ℤ_{p}G′, while v_{x} lies in ℤ_{p}xG′, the element u_{x} also lies in ℤ_{p}xG′, so is certainly distinct from E. We need to prove that u_{x} ≠ E in general.

For this purpose, it now suffices to consider the case that G = G′, so that p^{3} divides |G|. When r = 1, we know that K_{x}^{p} − K_{x^{2}}^{p} lies in Tr_{1}^{G}(ℤG), so that

the coefficient of 1_{G} in ^{\frac{K_{x}^{p}−K_{x^{2}}^{p}}{p}} is divisible by p.

When r > 1, the coefficient of 1_{G} in (K_{x^{2}}^{p} − K_{x}^{p})^{r} is

∑_{\{χ∈Irr(G):χ(x)≠0,p^{2}| \frac{|G|}{χ(1)} \}} (\frac{|G|}{χ(1)})^{r} \frac{χ(1)^{2}}{|G|} .

For χ of defect 3 or more, the contribution to the sum from χ is divisible by p^{3r−3}, so is divisible by p^{2r−1}. For χ of defect 2, we note that χ(1) is divisible by p, so that the contribution to the sum from χ is divisible by p^{2r−1}. Now for r ≥ 2, we know that 2r − 1 > r + \frac{r−σ_{p}(r)}{p−1}, so that the coefficient of 1_{G}

in ( r^{\frac{1}{p}} ) (K_{x^{2}}^{p} − K_{x}^{p})^{r} is divisible by p. Since K_{x^{2}}^{p} is a unit in Eℤ_{p}G, whose

image under reduction (mod p) composed with Br_{⟨x⟩} is 1, and since \frac{K_{x^{2}}}{K_{x}} is a unit in Eℤ_{p}G, whose image under reduction (mod p) composed with Br_{⟨x⟩} is x, we may conclude (using the isolation property) that the coefficient of x in u_{x} is congruent to 1 ( mod p).

REMARK: We remark that if λ, μ are elements of ℤ_{p}[η]\(1 − η)ℤ_{p}[η], and λ^{p} − μ^{p} lies in (1 − η)^{m}ℤ_{p}[η]\(1 − η)^{m+1}ℤ_{p}[η], for some m > 0, then m ≥ p, and that if m > p, then there is a unique value of i with 0 ≤ i ≤ p − 1 such that λ − η^{i}μ ∈ (1 − η)^{m−(p−1)}.

Choose i with 1 < i < p. For each irreducible character χ of G lying in a block of defect greater than 1, with χ(x) ≠ 0, we conclude, since ω_{χ}(K_{x^{i}})^{p} ≡ ω_{χ}(K_{x})^{p} ( mod \frac{|G|_{p}}{χ(1)_{p}}), that there is a unique integer m_{χ}(i) with 0 ≤ m_{χ}(i) ≤ p − 1 and

ω_{χ}(K_{x^{i}}) ≡ η^{m_{χ}(i)}ω_{χ}(K_{x}) ( mod \frac{|G|_{p}}{pχ(1)_{p}}). On the other hand, setting

ω_{χ}(K_{x}) = ∑_{j=1}^{p−1} m_{j}η^{j}

for (uniquely determined) integers m_{j}, we see that

ω_{χ}(K_{x^{i}}) − ω_{χ}(K_{x}) ≡ ∑_{j=1}^{p−1} jm_{j}(i − 1)(η − 1)(mod(1 − η)^{2}).

It follows easily that

m_{χ}(i)ω_{χ}(K_{x}) ≡ (i − 1) ∑_{j=1}^{p−1} jm_{j}(mod(1 − η)),

so that m_{χ}(i) ≡ (i − 1) (∑_{j=1}^{p−1} jm_{j}) ( mod p), as ω_{χ}(K_{x}) ≡ 1 ( mod (1 − η)).

We define the linear character λ_{χ} of ⟨x⟩ via λ_{χ}(x) = η^{∑_{j=1}^{p−1} jm_{j}}. The given series expansion for u_{x} shows that

ω_{χ}(u_{x}) ≡ η^{∑_{j=1}^{p−1} jm_{j}}(mod(1 − η)^{2}),

so that ω_{χ}(u_{x}) = η^{∑_{j=1}^{p−1} jm_{j}}, as ω_{χ}(u_{x}) is a power of η. Hence λ_{χ}(x^{i}) = ω_{χ}(u_{x})^{i} for 0 ≤ i ≤ p − 1.

Using the description above, the linear characters λ_{χ} behave well with respect to Galois conjugation (that is, if τ ∈ Gal(ℚ[η]/ℚ), then λ_{χ^{τ}} = (λ_{χ})^{τ} whenever χ is an irreducible character of G which does not vanish at x).

LEMMA 6: Let χ be an irreducible character of G with χ(x) ≠ 0 and let θ be a primitive |G|_{p′}-th root of unity. Then:

i) There is an integer t_{χ} such that χ(x) ≡ t_{χ}λ_{χ}(x) ( mod \frac{|G|_{p}}{p} ℤ[η]). ii) For each p-regular y ∈ H, we have

χ(xy) ≡ λ_{χ}(x)t_{χ,y}(mod\frac{|C_{G}(xy)|}{p}ℤ[η, θ]),

for some t_{χ,y} ∈ ℤ[θ]).

PROOF: i) We may, and do, suppose that χ does not lie in a p-block of defect 1. For each linear character μ of x, (recalling that (for 1 ≤ i ≤ p − 1) ω_{χ}(K_{x^{i}}) ≡ λ_{χ}(x^{i−1})ω_{χ}(K_{x}) ( mod \frac{|G|_{p}}{pχ(1)_{p}})), we see that

χ(x) ≡ λ_{χ}(x)t_{χ}(mod\frac{|G|_{p}}{p} ℤ[η])

for the rational integer

t_{χ} = χ(1) − p⟨Res_{⟨x⟩}^{G}(χ), μ⟩,

where μ is any linear character other than λ_{χ} of ⟨x⟩.

Let us write χ(x) = λ_{χ}(x)t_{χ} + \frac{|G|_{p}}{p} α for some α ∈ ℤ[η]. Since we know that

\frac{χ(x)}{χ(1)} ∈ ℤ_{p}[η]\(1 − η)ℤ_{p}[η],

we easily deduce that t_{χ} and χ(1) are exactly divisible by the same power of p.

Choose any p-regular y ∈ H. Then y^{−1}x^{−1}x^{g} is (by the minimality of G) p-regular whenever ⟨xy, x^{g}⟩ ≠ G, and it follows ( as in [4]) that

\frac{[G : H]χ(x^{−1})χ(xy)}{χ(1)} = γ_{y} + |C_{G}(xy)|_{p}δ

for some γ_{y} ∈ ℤ[θ] and some algebraic integer δ. We easily obtain ( recalling that \frac{[G:C_{G}(xy)]χ(xy)}{χ(1)} is an algebraic integer):

χ(xy) ≡ λ_{χ}(x)t_{χ,y} ( mod \frac{|C_{G}(xy)|_{p}}{p}) for some t_{χ,y} ∈ ℤ[θ].

Let us now exploit these congruences. First, we recall some standard facts (and notation). For a finite group L and a p-element y of L, we let S_{p}^{L}(y) denote the p-section of y in L ( that is, the set of elements of L whose p-parts are L-conjugate to y). We recall that induction of class functions gives an isometry from the space of complex-valued class functions of C_{L}(y) which vanish identically off S_{p}^{C_{L}(y)}(y) to the space of complex-valued class functions of C_{L}(y) which vanish identically off S_{p}^{L}(y), and that the induced class function agrees with the original class function on S_{p}^{C_{L}(y)}(y). We recall also a well-known consequence of Brauer’s characterization of characters: a virtual Brauer character β of the finite group L extends to a generalized character vanishing identically off p-singular elements if and only if \frac{β(w)}{|C_{L}(w)|_{p}} is an algebraic integer for each p-regular w ∈ L.

Consider χ as in Lemma 6. For 0 ≤ i ≤ p − 1, let ϕ_{i} denote the character of H associated to its action on the η^{i}-eigenspace of x ( in any representation affording χ). Let λ_{χ}(x) = η^{j}.

Consider first the case that j ≠ 0. Then, replacing η by an algebraic conjugate if necessary, we may, and do, assume that j = 1. Let ψ_{i} denote the virtual Brauer character associated to ϕ_{i} − ϕ_{0}, which we may regard as a virtual Brauer character for H/⟨x⟩. Let σ be a Galois automorphism which fixes p′-roots of unity and acts as an automorphism of order p − 1 of ℚ[η].

Then Res_{H}^{G}(χ) agrees with ∑_{i=1}^{p−1} λ_{χ}^{i} ∗ ψ_{i} on elements of H whose p-parts lie in ⟨x⟩^{#}, where, for each i, λ_{χ}^{i} ∗ ψ_{i}(x^{j}y) = η^{ij}ψ_{i}(y) for 1 ≤ j ≤ p − 1 and p-regular y ∈ H, and λ_{χ}^{i} ∗ ψ_{i} vanishes on elements of H not of this form. The congruences achieved in Lemma 6 tell us that ψ_{i} is the Brauer character of a virtual projective RH/⟨x⟩-module for i ≠ 1. By Brauer’s Second Main Theorem, each ψ_{i} is in the ℤ-span of Brauer characters in the unique block of RH which is the Brauer correspondent of the block of RG containing χ. Since this block does not have defect 1, we note that it contains no irreducible ⟨x⟩-projective RH-module.

By ( for example), the results of Külshammer-Robinson [2] and Navarro [3], the class functions λ_{χ}^{i} ∗ ψ_{i} are mutually orthogonal (and are generalized characters when i ≠ 1). However, λ_{χ} ∗ ψ_{1} need not be a generalized character of H.

For ease of exposition, let us write ( on elements of H whose p-parts are non-identity elements of ⟨x⟩), that Res_{H}^{G}(χ) agrees with ∑_{i=0}^{p−2} λ_{χ}^{σ^{i}} ∗ Δ_{i}. Then we see that Res_{H}^{G}(χ − χ^{σ}) agrees with

λ_{χ} ∗ (Δ_{0} − Δ_{p−2}) + ∑_{i=1}^{p−2} λ_{χ}^{σ^{i}} ∗ (Δ_{i} − Δ_{i−1})

on all elements of H whose p-parts are powers of x ( including 1_{H}). Now for i > 1, we see that λ_{χ}^{σ^{i}} ∗ (Δ_{i} − Δ_{i−1}) is a generalized character of H, and we notice that also

∥λ_{χ} ∗ (Δ_{0} − Δ_{p−2})∥^{2} + ∑_{i=1}^{p−2} ∥λ_{χ}^{σ^{i}} ∗ (Δ_{i} − Δ_{i−1})∥^{2} ≤ 2.

We claim that this forces Δ_{i}−Δ_{i−1} = 0 for all i > 1. Otherwise ∥Δ_{i}−Δ_{i−1}∥^{2} = 2 for exactly one value of i > 1 ( as the relevant block of RH is not of defect 1, so contains no block of defect zero of H/⟨x⟩), and then we have (Δ_{0} − Δ_{p−2}) = 0 and (Δ_{j} − Δ_{j−1}) = 0 for all j ∉ {0, i}. This is impossible as clearly

(Δ_{0} − Δ_{p−2}) + ∑_{j=1}^{p−2}(Δ_{j} − Δ_{j−1}) = 0.

Hence we do indeed have Δ_{i}−Δ_{i−1} = 0 for all i > 1, so that Δ_{1} = Δ_{2} = … Δ_{p−2} and Res_{H}^{G}(χ) agrees with

λ_{χ} ∗ Δ_{0} + ∑_{i=1}^{p−2} λ_{χ}^{σ^{i}} ∗ Δ_{1}

on p-sections (in H) of non-identity powers of x. Expressed differently, we note that Res_{H}^{G}(χ) agrees with

λ_{χ} ∗ (Δ_{0} − Δ_{1}) − (1 ∗ Δ_{1})

on p-sections (in H) of non-identity powers of x. Notice that the latter generalized character agrees with Δ_{0} − 2Δ_{1} on p-regular elements of H and vanishes outside p-sections of (possibly identity) powers of x.

Since λ_{χ} ∗ (Δ_{0} − Δ_{1}) and (1 ∗ Δ_{1}) are orthogonal class functions of H, we deduce that

\frac{1}{|G|} ( ∑_{u∈∪_{j=1}^{p−1}S_{p}^{G}(x^{j})} |χ(u)|^{2}) = \frac{1}{|H|} ( ∑_{u∈∪_{j=1}^{p−1}S_{p}^{H}(x^{j})} |χ(u)|^{2})

= ∥Δ_{0} − Δ_{1}∥^{2} + ∥Δ_{1}∥^{2} − \frac{1}{p} (∥Δ_{0} − 2Δ_{1}∥^{2}) .

Since χ is irreducible and does not vanish at the identity, this quantity is strictly less than 1. It also has the form

∥v∥^{2} + ∥w∥^{2} − \frac{∥v + w∥^{2}}{p} ,

where {v, w} = {(Δ_{0} − Δ_{1}), −Δ_{1}}. We leave a choice deliberately, since we suppose in what follows that ∥v∥ ≥ ∥w∥.

Now

(∥v∥^{2} + ∥w∥^{2} − \frac{∥v + w∥^{2}}{p} ) ≥ \frac{(p − 2)(∥v∥^{2} + ∥w∥^{2}) + (∥v∥ − ∥w∥)^{2}}{p} .

Now set ∥w∥ = t∥v∥, where 0 < t < 1. Then

\frac{(p − 2)(∥v∥^{2} + ∥w∥^{2}) + (∥v∥ − ∥w∥)^{2}}{p} = \frac{∥v∥^{2}}{p} ((p − 2)(1 + t^{2}) + (1 − t)^{2}) .

Now ((p − 2)(1 + t^{2}) + (1 − t)^{2}) takes its minimum value when t = \frac{1}{p−1}, and

the value taken there is \frac{p(p−2)}{p−1}.

Hence we have

1 > (∥Δ_{0} − Δ_{1}∥^{2} + ∥Δ_{1}∥^{2} − \frac{1}{p}∥Δ_{0} − 2Δ_{1}∥^{2}) ≥ \frac{(p − 2)max\{∥Δ_{0} − Δ_{1}∥^{2}, ∥Δ_{1}∥^{2}\}}{p − 1}.

This forces Δ_{1} = 0, since otherwise ∥Δ_{1}∥^{2} ≥ 2, as Δ_{1} is the virtual character of a virtual projective of H/⟨x⟩, but its irreducible constituents lie in a block of defect greater than 1 of H.

We are left to consider the case that λ_{χ} is the trivial character. Let σ be the Galois automorphism above. Then, using the congruences of Lemma 6 again, we see that Res_{H}^{G}(χ) agrees with

1 ∗ Λ_{0} + ∑_{i=1}^{p−2} λ_{χ}^{σ^{i}} ∗ Λ_{i}

on ∪_{j=1}^{p−1}S_{p}^{H}(x^{j}), where each Λ_{i} is a virtual Brauer character of H/⟨x⟩, and is virtual projective for i > 0. Furthermore, each Λ_{i} lies in the unique Brauer correspondent for H of the block of RG containing χ, so lies in a block of defect greater than 1 of H. This time, Res_{H}^{G}(χ − χ^{σ}) agrees with

λ^{σ} ∗ Λ_{1} − λ ∗ Λ_{p−2} + ∑_{i=2}^{p−2} λ_{χ}^{σ^{i}} ∗ (Λ_{i} − Λ_{i−1})

on ∪_{j=0}^{p−1}S_{p}^{H}(x^{j}). Each of the p−1 virtual characters appearing here is the virtual character of an ⟨x⟩-projective RH-module, and they are mutually orthogonal. Since (χ − χ^{σ}) has norm-squared 0 or 2, and since none of the virtual characters appearing has norm 1, we must conclude that at most one of the virtual characters Λ_{1}, Λ_{2} − Λ_{1}, …, Λ_{p−2} − Λ_{p−3}, −Λ_{p−2} is non-zero. Since these virtual characters sum to zero, they must all be zero. Hence Res_{H}^{G}(χ) agrees with 1∗Λ_{0} on ∪_{j=1}^{p−1}S_{p}^{H}(x^{j}) in this case.

Notice that we have proved along the way that λ_{χ} is the trivial character if and only if χ(x) is rational. Before we state a Theorem collecting some consequences of these observations, we produce the proof of the following wellknown fact for the convenience of the reader.

LEMMA 7: Let X ∈ M_{r}(ℤ_{p}) be a matrix of order p such that (X − I)^{n} ∈ pM_{r}(ℤ_{p}) for some n < (p − 1). Then X = I.

PROOF: Let us regard X as a matrix in X ∈ M_{r}(ℤ_{p}[η]). If X ≠ I, then there is an r-long column vector v with entries in ℤ_{p}[η], but not all entries in (1 − η)ℤ_{p}[η] such that Xv = ηv. Then (X − I)^{n}v = (η − 1)^{n}v has at least one of its entries outside (η − 1)^{n+1}ℤ_{p}[η], so outside pℤ_{p}[η], contrary to the fact that (X − I)^{n} has all entries divisible by p.

Now we prove:

THEOREM 2: i) The unit u_{x} is equal to \frac{K_{x^{2}}}{K_{x}} .

ii) In the case that G = G′ and p > 3, then there is a bijection between irreducible characters of G whose values at x are irrational and irreducible characters of H which do not have x in their kernel. PROOF: Whenever χ is an irreducible character of G in a p-block whose defect group is ⟨x⟩, we know that ω_{χ} (\frac{K_{x^{2}}}{K_{x}} ) is a power of η. If χ is an irreducible character in a block of RG of defect greater than one whose defect group contains x, then we have seen above that ω_{χ} (\frac{K_{x^{2}}}{K_{x}} ) = λ_{χ}(x).

ii)We prove that if p > 3, then Z(Eℤ_{p}G) is a permutation module for ⟨u_{x}⟩ (under right translation). We will see that the rank of the fixed-point submodule is the number of irreducible characters of G whose value at x is a non-zero rational integer and that the rank of the quotient module by this fixed-point submodule is k(H) − k(H/⟨x⟩), which is the number of irreducible characters of H taking non-rational values at x.

Let U = Z(Eℤ_{p}G), T = Tr_{1}^{G}(Eℤ_{p}G). Now the Brauer homomorphism Br_{⟨x⟩} shows that u_{x}^{∗} acts on U^{∗}/T^{∗} as x does on Z(GF(p)H), since u_{x}^{∗}Br_{⟨x⟩} = x. Let x have r projective indecomposable summands on the latter module. Then we may write U^{∗} = P^{∗} ⊕ Q^{∗}, where P^{∗} is a free GF(p)⟨u_{x}^{∗}⟩-module of GF(p)-dimension pr which has trivial intersection with T^{∗}. Now U^{∗}(E^{∗} −u_{x}^{∗}) ⊆ P^{∗}(E^{∗} −u_{x}^{∗})+T^{∗} on consideration of dimension. However, as u_{x}^{∗} − E^{∗} is nilpotent, it annihilates T^{∗}. Thus U^{∗}(E^{∗} − u_{x}^{∗})^{2} = P^{∗}(E^{∗} − u_{x}^{∗})^{2} and Q^{∗}(E^{∗} − u_{x}^{∗})^{2} = {0}. Let P be a projective pre-image of P^{∗} which occurs as a summand of U (as ℤ_{p}⟨u_{x}⟩-module), and let V be a complement to P in U. By Lemma 7 ( since p > 3), we see that u_{x} acts trivially on V (in fact, this is the only place where we use the fact that p > 3).

Extending scalars to 𝕂, the primitive central idempotents e_{χ} for which χ(x) ≠ 0 form a u_{x}-invariant basis for Z(E𝕂G) and u_{x}e_{χ} = e_{χ} ( for such χ) if and only if χ(x) is rational. Hence the rank of the (ℤ_{p}-pure) submodule of Z(Eℤ_{p}G) consisting of u_{x}-fixed points is as stated. The ℤ_{p}-rank of the quotient module is p − 1 times the number of indecomposable projective summands of Z(Eℤ_{p}G) as ℤ_{p}⟨u_{x}⟩-module, which (as noted above) is also p − 1 times the number of projective summands of Z(GF(p)H) as ⟨x⟩-module under right multiplication. On the other hand, this is the number of irreducible characters of H whose value at x is not rational, by an argument like that above.

ACKNOWLEDGEMENT: We thank the referee for reading the paper very carefully, and suggesting some shorter arguments, especially in Lemma 3. We also acknowledge a debt to M.J. Collins for originally introducing us to this problem.

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