BOUNDING NUMBERS AND HEIGHTS OF CHARACTERS IN

p-CONSTRAINED GROUPS Geoffrey R. Robinson School of Mathematics and Statistics University of Birmingham Birmingham B15 2TT England

INTRODUCTION: Positive confirmation of Brauer’s k(B)-problem ( which is to prove that the number of ordinary irreducible characters in a p-block B is at most |D|, where D is a defect group for B ) has recently been achieved in the case that B is a block of a p-solvable group, which represents the culmination of the work of many authors, beginning with Nagao [6] and Knörr [4]. By contrast, the k(B) problem for p-constrained groups remains open at present.

In this article, we present a new character-theoretic conjecture for p-con-strained groups, which is, for p-solvable groups, strictly stronger than the k(B)- problem. The new conjecture seems neither to imply, nor to be implied by, the k(B)-problem for the general p-constrained group. In some ways, though, it is more precise for such a group. A key observation is Theorem 1, which may be of independent interest, and which clears the way for the character-theoretic methods successfully employed in the k(GV)-problem to be used in more general situations. In particular, we prove that the new conjecture holds in fairly general circumstances, to be described more precisely later. The generalized characters constructed in Theorem 5 may also be of independent interest.

SECTION 1: THE CONJECTURE AND PROOF IN SPECIAL CASES

We consider a finite group G which has a non-trivial Abelian normal (p)- subgroup V such that C_{G}(V) is a p-group. Let S be a Sylow p-subgroup of G, and set P = S/V. For each v ∈ C_{V} (S), set H(v) = C_{G}(v)/V and let H(v)_{p′} denote the set of p-regular elements of H(v). We abbreviate H(1) to H.

We let k_{d}(G) is the number of irreducible characters, χ, of G which satisfy p^{d}χ(1)_{p} = |G|_{p}. We make the following:

CONJECTURE: Under the hypotheses above, we have

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ max\{\frac{|H(v)_{p′}|}{|V ||H(v)|} : v ∈ C_{V} (S)\}.

In particular,

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ \frac{1}{|V |},

with equality only possible if V ∈ Syl_{p}(G).

REMARKS: It would be of some interest to know whether

max\{\frac{|H(v)_{p′}|}{|H(v)|} : v ∈ C_{V} (S)\} = \frac{|H_{p′}|}{|H|},

as this would allow a cleaner statement of the conjecture, but we are unsure whether this is the case in general.

We will henceforth refer to the second version of the inequality as the “weak version” of the conjecture. Note that it asserts that a necessary condition for equality is that V ∈ Syl_{p}(G), but does not assert that this is sufficient to yield equality.

Another interpretation of the conjecture ( in the weak form, for ease of exposition), is that

∑_{χ∈Irr(G)} χ(1)_{p}^{2} ≤ \frac{|G|_{p}^{2}}{|V |} .

For purposes of comparison, the reader might like to note that ( after reconciling notation), it is a consequence of Brauer-Feit [1] that

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ 1,

and (implicitly) that

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ \frac{|G_{p′}|}{|G|}.

While our conjectured bound may at first glance look only marginally sharper, the difference is significant. For example, if we additionally assume that G/V has a normal p-complement, then ( the stronger version of) our conjectured bound ( suitably re-written) translates to

∑_{μ∈Irr(G)} μ(1)_{p}^{2} ≤ |G|_{p}.

This serves to illustrate that our conjectured bound is sometimes sharper that the bound predicted in Brauer’s k(B)-problem, which in the last example would reduce to k(G) ≤ |G|_{p}. It also shows that the new conjecture ( even in its weak form) implies the k(GV)-conjecture, and hence Brauer’s k(B) problem for psolvable groups.

In fact, in conjunction with the solution by Gluck and Wolf in [2] of Brauer’s height zero conjecture for p-solvable groups and Nagao’s fundamental inequality k(X) ≤ k(Y)k(X/Y) whenever Y ⊲ X, the new conjecture implies that whenever L is a p-solvable group with O_{p′}(L) = 1 which is not p-closed, we have k(L) < |L|_{p}. For we may assume that O_{p′}(L/M) ≠ 1 whenever M is a minimal normal subgroup of L. In particular, this implies that Φ(L) = 1. Now take T = O_{p,p′,p}(L). Then O_{p}(T) is elementary Abelian, but T has non-Abelian Sylow p-subgroups, so T has an irreducible character of degree divisible by p. Since we saw above that the conjecture yields

∑_{μ∈Irr(T)} μ(1)_{p}^{2} ≤ |T |_{p},

it certainly yields k(T) < |T |_{p}, while also k(L/T) ≤ [L : T]_{p}.

Another consequence of the weak form of our conjecture would be that G/V should have at most |V | p-blocks of defect 0, or, more precisely, that G should have at most |V | irreducible characters which are V -projective (an irreducible character χ is V -projective if and only if its degree is divisible by [G : V]_{p}). Notice that the weaker version of our conjecture would be sure to hold for G if we knew that k(G) ≤ |V |, since the degree of any irreducible character of G divides [G : V] by Ito’s Theorem. However, it is not generally true that k(G) ≤ |V | under our hypotheses.

We also note that the inequality

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ \frac{1}{|G|_{p}}

does not hold in general when G is p-solvable with O_{p′}(G) = 1, as the example G = GL(2, 3) already illustrates ( for p = 2). We finally remark that the case of p = 3 and G the Hessian group of order 3^{4}2^{3} ( recall that G is the semi-direct product on an extra-special group of order 27 and exponent 3 with SL(2, 3), where the action is the one obtained naturally from the fact that SL(2, 3) = Sp(2, 3))) shows that the inequality

∑_{d=0}^{∞} \frac{k_{d}(G)}{p^{2d}} ≤ \frac{1}{|O_{p}(G)|}

can fail even for p-solvable G ( with O_{p′}(G) = 1), when O_{p}(G) is non-Abelian. Hence the assumption that V is Abelian is crucial.

NOTATION: If a finite group X acts on a p-group U, we will say that the Brauer character of X afforded by U is that afforded by the action of X on the direct sum of the chief factors of XU contained within U, viewed as GF (p)Xmodules.

We will prove that the conjecture holds in the case that the Brauer character afforded by V is real-valued, or more generally, in the case that there is an element v ∈ C_{V} (S) such that Res_{C_{G}(v)}^{G}(V) has a submodule U on which no non-identity p-regular element acts trivially, and which affords a real-valued Brauer character. These are similiar to the circumstances under which the k(GV)-conjecture was shown to hold in [8] ( where S = V), though the proof is in some ways more delicate in this case.

We also recall some general facts which are useful, and will be freely used in what follows:

i) If X is a finite group with a Sylow p-subgroup Q, and ϕ is a Brauer character ( for the prime p) of X, then the class function of X which vanishes on all psingular elements, and agrees with |P |ϕ on all p-regular elements, is a generalized character. ii) X is as above, and X has a normal p-subgroup W, then there is a natural bijection between generalized characters of X which vanish on all p-singular elements and generalized characters of X/W which vanish on all p-singular elements. In this bijection, the generalized character θ of X corresponds to the generalized character θ^{∗} of X/W which satisfies

θ^{∗}(xW) = \frac{θ(x)}{|C_{W} (x)|}

for all p-regular x ∈ X.

The first assertion above is well-known. The second is frequently used in [8], and is due to W.F. Reynolds.

The next auxiliary result is true in complete generality.

THEOREM 1: Let X be a finite group, A be an Abelian normal subgroup of X, and let μ be a complex irreducible character of X. Then for every y ∈ X,

\frac{[X : C_{X}(y)]μ(y)}{[A : C_{A}(y)]μ(1)}

is an algebraic integer.

PROOF: Let ω = exp(\frac{2πi}{|X|}) and let R = ℤ[ω]_{ρ} be the localization at some prime ideal ρ of ℤ[ω]. Since ρ is arbitrary, it suffices to prove that

\frac{[X : C_{X}(y)]μ(y)}{[A : C_{A}(y)]μ(1)} ∈ R.

Let λ be a linear constituent of Res_{A}^{X}(μ), and let H = I_{X}(λ). We may write μ = Ind_{H}^{X}(ζ) for some irreducible character ζ of H which lies over λ. Let S^{+} denote the sum of the elements of S in RX, whenever S is a subset of X.

A routine calculation shows that

\frac{μ((y^{X})^{+})}{μ(1)} = \frac{ζ((y^{X} ∩ H)^{+})}{ζ(1)},

so it suffices by induction to consider the case that H = X, since y^{X} ∩ H is a union of conjugacy classes of H, and since [A : C_{A}(y′)] = [A : C_{A}(y)] whenever y′ is an X-conjugate of y. Hence, we may, and do, suppose that λ is X-stable. Let V be an RX-module affording character μ, and let σ : X → End_{R}(V) be the associated representation. We view E = End_{R}(V) as an interior X-algebra in the usual fashion, employing σ.

We note that A acts trivially on E, since elements of A act as scalars on V by the stability of λ. Hence we see that

T r_{C_{X}(y)}^{X}(yσ) = [A : C_{A}(y)]T r_{AC_{X}(y)}^{X}(yσ),

while T r_{AC_{X}(y)}^{X}(yσ) is a scalar matrix with entries in R. From this, we conclude that

\frac{[X : C_{X}(y)]μ(y)}{[A : C_{A}(y)]μ(1)} ∈ R,

as required.

Now let us return to our particular situation. If p is an odd prime, we let ε be the sign for which p ≡ ε ( mod 4), and we let R be the ring of algebraic integers of ℚ[\sqrt{εp}]. If p = 2, we let R = ℤ.

A key result for the cases of the conjecture we can prove, which is related to Knörr’s notion in [4], of “containing a square” is:

LEMMA 2: Let K be a finite group which acts on a p-group U in such a way that U affords a real-valued Brauer character of K. Then there is an Rcombination of Brauer characters, θ, of K such that:

i) |θ(x)|^{2} = [U : C_{U}(x)] for each p-regular x ∈ K. ii) θ is integer valued if p = 2, and for p odd, θ(x) has the form ε(x)(\sqrt{εp})^{m(x)} for some sign ε(x) and some integer m(x), ( for each x ∈ K).

PROOF: It suffices to consider the case that U is elementary Abelian and completely reducible as GF (p)K-module. In that case, by the results of [8], which rely on the results of Gow [3], we can find an integer-valued Brauer character ψ of K such that either:

i) ψ(x)^{2} = [U : C_{U}(x)] for all p-regular x ∈ K. or ii)ψ(x)^{2} = [U : C_{U}(x)] for all p-regular x ∈ K which act with determinant 1 on U, and ψ(x)^{2} = \frac{[U:C_{U} (x)]}{p} for all p-regular x ∈ K which act with determinant -1 on U.

In the second case ( which only occurs when p is odd), let L be the normal subgroup of K consisting of those elements which acts with determinant 1 on U. Let χ be the class function which takes the value 1 on L and \sqrt{εp} on K\L. Then χ = \frac{1+\sqrt{εp}}{2}1_{K} + \frac{1−\sqrt{εp}}{2}λ, where λ is the character afforded by the determinant on U. Hence χ is an R-combination of characters of K, so we may set θ(x) = χ(x)ψ(x) for all p-regular x ∈ K to obtain the desired conclusion.

Now let us turn to the proof of the conjecture in some special cases. Suppose first that C_{G}(V) is a p-group, and that V contains a normal subgroup U of G such that C_{G}(U) is a p-group and such that U affords a real-valued Brauer character.

We let R be as above. Let P be a Sylow p-subgroup of G/V. We have an R-combination of Brauer characters of G, say θ, such that |θ(x)|^{2} = [U : C_{U}(x)] for all p-regular x ∈ G. We may, and do, regard θ as an R-combination of Brauer characters of G/V. Then |P |θ is an R-combination of characters of projective indecomposable modules of G/V, and the class function,ψ, of G defined by ψ(x) = |C_{V} (x)||P |θ(x) for all p-regular x ∈ G is an R-combination of characters of projective indecomposable modules of G.

Let us compute ⟨ψ, μ⟩ for an irreducible character μ of G. We know from Theorem 1 that \frac{[G:C_{G}(x)]μ(x)}{μ(1)} is an algebraic integer multiple of [V : C_{V} (x)] for

each p-regular x ∈ G. Hence \frac{[G:C_{G}(x)]μ(x)ψ(x^{−1})}{μ(1)} is an algebraic integer multiple of |P ||V |θ(x^{−1}) for each p-regular x ∈ G. Now θ(1) = ±1, while θ(x^{−1}) is divisible by \sqrt{εp} ( or by 2 in the case p = 2) for each non-identity p-regular element x of G. Since |G|_{p} = |P ||V |, we deduce that ⟨ψ, μ⟩ is divisible ( in R) by μ(1)_{p} but not by \sqrt{εp}μ(1)_{p} ( or by 2μ(1)_{2} in case p = 2).

In the following calculation, the reader may find it helpful to recall that there is a bijection between the classes of p-regular elements of G and of G/V, and that if x is a p-regular element of G, then the number of conjugates of xV in H is \frac{[G:C_{G}(x)]}{[V:C_{V} (x)]}.

We have

|G|⟨ψ, ψ⟩ = ∑_{x∈G_{p′} /G}[G : C_{G}(x)][U : C_{U}(x)]|C_{V} (x)|^{2}|P |^{2}

≤ ∑_{x∈G_{p′} /G}[G : C_{G}(x)]|V ||C_{V} (x)||P |^{2},

so that

|H|⟨ψ, ψ⟩ ≤ ∑_{x∈G_{p′} /G} \frac{[G : C_{G}(x)]|V ||P |^{2}}{[V : C_{V} (x)]}

and hence

⟨ψ, ψ⟩ ≤ \frac{|V ||P |^{2}|H_{p′}|}{|H|}

Since

⟨ψ, ψ⟩ ≥ ∑_{μ∈Irr(G)} μ(1)_{p}^{2},

dividing both sides by |P |^{2}|V |^{2} gives the result. To be more precise, when p is odd, we see that

⟨ψ, ψ⟩ = ⟨ψ^{σ}, ψ^{σ}⟩

where σ is a Galois automorphism fixing p′-roots of unity and sending \sqrt{εp} to its negative), and we note that

|α^{σ}|^{2} + |α|^{2} ≥ 2

for any non-zero α ∈ R.

We have now dealt with the case that V affords real-valued Brauer character, or more generally, when it has a subgroup U which affords real-valued Brauer character and for which C_{G}(U) is a p-group.

Suppose now that there is some non-identity element v ∈ C_{V} (S) such that Res_{C_{G}(v)}^{G}(V) has a subgroup U with the same property with respect to C_{G}(v). We may define a class function δ analogously for C_{G}(v) so that |δ(x)|^{2} = [U : C_{U}(x)] for all p-regular x ∈ C_{G}(v).

We may, and do, regard δ as an R-combination of Brauer characters of H(v) = C_{G}(v)/V. Then |P |δ is an R-combination of characters of projective indecomposable modules of H(v), and the class function,ψ, of C_{G}(v) defined by ψ(x) = |C_{V} (x)||P |δ(x) for all p-regular x ∈ C_{G}(v) is an R-combination of characters of projective indecomposable modules of C_{G}(v).

Now ( letting ζ denote a primitive complex |v|-th root of unity), we define the ℤ[ζ]-combination of characters β of C_{G}(v) via β(vx) = ψ(x) if x is pregular, β(x) = 0 otherwise. W then consider the induced class function ψ = Ind_{C_{G}(v)}^{G}(β), which vanishes outside the p-section of v in G, and which has the same norm as β does.

Again, we know from Theorem 1 that whenever μ is an irreducible character of G, \frac{[G:C_{G}(xv)]μ(xv)}{μ(1)} is an algebraic integer multiple of [V : C_{V} (xv)] for each p-

regular x ∈ C_{G}(v). Hence \frac{[G:C_{G}(xv)]μ(xv)ψ((xv)^{−1})}{μ(1)} is an algebraic integer multiple of |P ||V |δ(x^{−1}) for each p-regular x ∈ C_{G}(v). Again, δ(1) = ±1, while δ(x^{−1}) is divisible by \sqrt{εp} ( or by 2 in the case p = 2) for each non-identity p-regular element x of G. Furthermore, \frac{[G:C_{G}(v)]μ(v)}{μ(1)} lies in ℤ[ζ]\(1 − ζ)ℤ[ζ], since μ is in the principal ( and only ) p-block of G, and v ∈ Z(S).

We deduce that ⟨ψ, μ⟩ is divisible ( in ℤ[ζ]) by μ(1)_{p} but not by (1−ζ)μ(1)_{p}. The computation showing that

⟨ψ, ψ⟩ ≤ \frac{|V ||P |^{2}|H(v)_{p′}|}{|H(v)|}

follows as before, using the fact that

⟨ψ, ψ⟩ = ⟨β, β⟩.

We now need to invoke a standard argument using algebraic conjugation ( essentially dating back to Burnside, and exploiting the arithmetic- geometric mean inequality in the case of totally positive algebraic integers). For every τ ∈ Gal(ℚ[ζ]/ℚ), we have

⟨ψ^{τ} , ψ^{τ} ⟩ ≤ \frac{|V ||P |^{2}|H(v)_{p′}|}{|H(v)|},

and ( for μ as above), we know that

\frac{⟨ψ^{τ} , μ(1)⟩}{μ(1)_{p}} ∈ ℤ[ζ]\(1 − ζ)ℤ[ζ].

Hence

∑_{τ∈Gal(ℚ[ζ]/ℚ)} |⟨ψ^{τ} , μ(1)⟩|^{2} ≥ |Gal(ℚ[ζ]/ℚ)|μ(1)_{p}^{2}.

Hence we may still conclude that

∑_{μ∈Irr(G)} μ(1)_{p}^{2} ≤ \frac{|V ||P |^{2}|H(v)_{p′}|}{|H(v)|},

which completes the proof that the strong form of the conjecture holds under the present hypotheses. REMARK: The interested reader may care to note that it follows from the proof of the conjecture in the cases above, that if H is a subgroup of S_{n} and G = ℤ_{2} ≀ H, then we have

∑_{χ∈Irr(G)} χ(1)_{2}^{2} ≤ 2^{3n−2}.

This already follows from the weak form of the conjecture, using the fact that a Sylow 2-subgroup of S_{n} has order at most 2^{n−1}. Notice in particular, that this yields

∑_{χ∈Irr(H)} χ(1)_{2}^{2} ≤ 2^{3n−2}.

SECTION 2: CLIFFORD-THEORETIC REDUCTIONS AND RESIDUAL OBSTACLES

The Clifford-theoretic reductions of the paper [8] and [9] carry over without any real difficulty to the present situation. We have seen that a sufficient condition for the conjecture to hold is the existence of an element v ∈ V with [G : C_{G}(v)] coprime to p such that Res_{C_{G}(v)}^{G}(V) has a subgroup U which affords a real Brauer character for C_{G}(v). In the situation of [8], the condition that [G : C_{G}(v)] is coprime to p is automatically satisfied, but in this new setting, we need to keep track of the condition. However, this is routine.

This means that one critical configuration which remains to consider after Clifford-theoretic reductions have been performed is the following: We have a finite group L which acts faithfully and absolutely irreducibly on a \overline{GF (p)}L-module U. L has a Sylow p-subgroup T, and there is no u ∈ U^{T} for which Res_{C_{L}(u)}^{L}(U) has a faithful submodule W which affords a real-valued Brauer character. Furthermore, we may suppose that either:

i) F^{∗}(L) = Z(L)O_{q}(L), where q ≠ p is a prime, and O_{q}(L) has all characteristic Abelian subgroups central. Furthermore, Q also acts absolutely irreducibly on U. ii) F^{∗}(L) = Z(L)E, where E is a quasi-simple subgroup acting absolutely irreducibly on U.

However, if E is a group of Lie type in characteristic p, for example, it is extremely likely that there will indeed be no such element u, since U^{T} is 1dimensional, so it seems that a different approach will be needed if such sections are involved in our group G.

Our next result is relevant to this “natural characteristic” situation, and is really concerned with the case when U as above is the natural module, which may be the most difficult case. Returning to our group G, let Λ be a set of representatives for the orbits of Irr(V) under the action of G, For a non-tivial λ ∈ Λ, let S_{λ} = I_{S}(λ), and assume ( as we may) that S_{λ} ∈ Syl_{p}(I_{G}(λ)). Let P_{λ} = S_{λ}/V. Suppose that I_{G}(λ)/V is still p-constrained ( in the strong sense that its largest normal p-subgroup contains its own centralizer). Let us count the sum of the squares of the p-parts of the degrees of those irreducible characters of G which lie over (G-conjugates of) λ.

This is [P : P_{λ}]^{2} times the corresponding sum for I_{G}(λ). Let V_{λ} be the kernel of λ. Then all irreducible characters of H_{λ} which lie over λ contain V_{λ} in their kernels and V/V_{λ} is cyclic and central in I_{G}(λ)/V_{λ}. Furthermore, λ may be identified with an I_{G}(λ)/V_{λ}-stable linear character of V/V_{λ}.

We wish to evaluate

∑_{χ∈Irr(I_{G}(λ),λ)} χ(1)_{p}^{2}.

We may replace I_{G}(λ) by H_{λ} = I_{G}(λ)/V_{λ} and set Z_{λ} = V/V_{λ}. By hypothesis, H_{λ} has only one p-block. From [7], we know that

∑_{χ∈Irr(H_{λ})} χ(1)_{p}^{2} ≤ \frac{|H_{λ}|_{p}^{2}}{|Z_{λ}|} .

We wish to count those characters which lie over a linear character of maximal order of Z_{λ}. We note that this contribution is independent of the algebraic conjugate chosen, and that there are at (p − 1)|Z_{λ}|/p such algebraic conjugates. Hence

∑_{χ∈Irr(I_{G}(λ),λ)} χ(1)_{p}^{2} ≤ \frac{p|P_{λ}|^{2}}{p − 1} .

We have thus proved:

THEOREM 4: Let G, V be as above and assume further that I_{G}(λ)/V is p-constrained for each non-trivial linear character λ of V. Then

∑_{χ∈Irr(G):V ≰ker(χ)} χ(1)_{p}^{2} ≤ \frac{p|P |^{2}(|Λ| − 1)}{p − 1},

where Λ is a set of representatives for the orbits of G on Irr(V).

REMARK: We remind the reader that the weak form of the conjecture asserts in this notation that we should have

∑_{χ∈Irr(G)} χ(1)_{p}^{2} ≤ |P |^{2}|V |,

so that under the hypotheses of the theorem above, we are some considerable way to achieving this conjectured inequality. In fact, if G/V is itself pconstrained, the inequality holds, using the Brauer-Feit bound. More generally, the inequality holds if G/V has at most |V | − \frac{p(|Λ|−1)}{p−1} p-blocks. In the case that G/V is a quasi-simple group of Lie type in characteristic p and V is its natural module, there are 1 + |Z(G/V)| p-blocks of G/V, and we certainly have |Z(G/V)| < (|V | − 1)^{\frac{1}{2}} , so the required inequality holds comfortably.

SECTION 3 : SOME GENERALIZED CHARACTERS

In this section, we note the existence of some generalized characters which are relevant to the questions considered earlier, but also have some independent interest. The relevance of these generalized characters lies in the fact that if the finite goup G acts faithfully on a GF (p)G-module U, we are interested in constructing a virtual Brauer character θ such that |θ(y)|^{2} ≤ [U : C_{U}(y)] for all p-regular y ∈ G, θ(y) ∈ π for all p-regular y ∈ G^{#}, and θ(1) ∉ π (where π is a suitable prime ideal containing the rational prime p).

THEOREM 5: Let X be a finite group, p be a prime, n be a positive integer relatively prime to p. Let U be a GF (p)G-module, and for each g ∈ X, let m_{n}(g, U) denote the number of primitive n-th roots of unity ( counting multiplicities) among the eigenvalues of g on U ⊗_{GF (p)} \overline{GF (p)}. Define a class function ψ_{n,U} of X via ψ_{n,U}(g) = p^{m_{g}(n,U)} for all g ∈ X. Then ψ_{n,U} is a generalized character of X.

PROOF: We note that ψ_{n,U} is constant on p′-sections, so it suffices to check that Res_{E}^{X}(ψ_{n,U}) is a generalized character for each Brauer elementary p′- subgroup, E, of X. Hence we may, and do, assume that X is a p′-group. We note that the result is visibly true when n = 1, since ψ_{1,U} is the permutation character afforded by the action of X on the vectors of U.

Let χ be the (Brauer) character of X afforded by U. Suppose that n > 1, and that p_{1}, p_{2}, …, p_{m} are the distinct prime divisors of n. Let J = {1, 2, …, m}. For each subset I ⊆ J, set p_{I} = ∏_{i∈I} p_{i}, where the empty product is defined as 1.

It is easy to check that for every g ∈ X, we have

m_{g}(n, U) = ∑_{I⊆J}(−1)^{|I|}⟨Res_{⟨g^{\frac{n}{p_{I}}} ⟩}^{X}(χ), 1⟩,

so that

ψ_{n,U}(g) = \frac{∏_{|I|even} |C_{U}(g^{\frac{n}{p_{I}}} )|}{∏_{|I|odd} |C_{U}(g^{\frac{n}{p_{I}}} )|}.

Using so-called Adams operations, we know that for each integer r, the class function for which g → |C_{U}(g^{r})| for all g ∈ X is a generalized character.

Hence ψ_{n,U} is an integer-valued ratio of generalized characters whose values are coprime to |X|, so is a generalized character. This last fact is a result of G. Mason [5], but we include a proof of a slightly stronger version of Mason’s result for the convenience of the reader:

LEMMA 6: Let X be a finite group, γ be an algebraic integer-valued class function of X with the property that γ(g^{n}) = γ(g)^{σ_{n}} whenever n is an integer coprime to |X| and σ_{n} is a Galois automorphism sending |X|-th roots of unity to their n-th powers. Suppose that there is a generalized character β of X such that β.γ is a generalized character of X and such that ∏_{g∈X:γ(g)≠0} β(g) is coprime to |X| ( this quantity is an integer under the current hypotheses ). Then γ is a generalized character.

PROOF: We may multiply β by various algebraic conjugates if necessary, and assume that β is integer-valued. We may find a positive integer r so that β(g)^{r} ≡ 1 ( mod |X|) whenever γ(g) ≠ 0. Hence we may replace β by β^{r} if necessary, and assume that β(g) ≡ 1 ( mod |X|) whenever γ(g) ≠ 0. We now have that \frac{γ(g)β(g)−γ(g)}{|X|} is an algebraic integer for all g ∈ X. It follows that ⟨β.γ − γ, χ⟩ is an algebraic integer whenever χ is an irreducible character of X. Since our hypotheses together imply that ⟨β.γ, χ⟩ ∈ ℤ and that ⟨γ, χ⟩ ∈ ℚ, we deduce that ⟨γ, χ⟩ ∈ ℤ. Hence γ is a generalized character, as χ was arbitrary.

REMARK: We note that ∏_{n>1} ψ_{n,U} takes the value [U : C_{U}(x)] for all pregular x ∈ X. We also remark that it is not really essential that U should be a module over the prime field to construct the generalized characters above, but that this is no real loss of generality in any case ( first, it is no loss of generality to assume that U is irreducible. In that case, the argument works for any finite field of characteristic p and size q, as long as we replace p^{m_{g}(n,U)} by q^{m_{g}(n,U)}. However, the same generalized character can clearly be constructed from an irreducible GF (p)X-module).

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