More Cancellation Theorems for Conjectures of Alperin and Dade Geoffrey R. Robinson ^{1} School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England e-mail : g.r.robinson@bham.ac.uk Dedicated to the memory of my cat Jimi

INTRODUCTION: In this paper, we give a number of conditions which ensure that chains in N (G, U)/N_{G}(U) ( U a p-subgroup of G, notation as in [7]), make zero contribution to the alternating sums which appear in the formulae of Dade’s Projective Conjecture ( DPC). As well as having some theoretical interest, when interpreted as making new predictions about control of character defects for groups in which DPC can be shown to hold in every section, such observations can simplify computations in particular examples. We also point out ( in a more precise way, only outlined here) that it would follow from DPC that the same consequences of Broué’s conjecture for numbers of irreducible characters in blocks with Abelian defect groups must still hold in p-power central extensions ( where characters of non-zero defect ( positive height) may occur), and for blocks covering such blocks of normal subgroups ( but which need not themselves have Abelian defect group).

REVIEW OF COPRIME AUTOMORPHISMS OF p-GROUPS

Here we collect together some known facts about p-groups and their automorphisms ( especially coprime ones) which will be useful to us later. Most of the results needed can be found in Gorenstein [3].

1This work was done while the author was a professeur invité at the University of Paris VII Denis Diderot. He is grateful for the support and hospitality received, and for the facilities extended to him by the Institut Henri Poincaré

Let G be a finite group, and H be a characteristic section of G ( that is, H has the form L/K where L and K are both characteristic subgroups of G ). We say that H is coprime automorphism preserving ( henceforth abbreviated to CAP) for G if C_{A}(H) = 1_{A} whenever A is a group of coprime automorphisms of G. Notice that if the section H is CAP for G and the section K is CAP for H, then the section K is CAP for G.

The following types of sections are known to be CAP for G :

i) Any characteristic subgroup H of G for which C_{G}(H) = Z(H) ( in particular, F^{∗}(G) is CAP for G , and if G is a p-group, then the Thompson subgroup J(G) is CAP for G). ii) If G is a p-group, then Ω(G) is CAP for G, where Ω(G) = Ω_{1}(G) if p is odd or if G is Abelian, and Ω(G) = Ω_{2}(G) if p = 2 and G is non-Abelian (as usual, Ω_{i}(G) = ⟨x ∈ G : x^{p^{i}} = 1_{G}⟩). iii) If G is nilpotent, then G/Φ(G) is CAP for G.

Somewhat more subtle CAP subgroups of p-groups were discovered and exploited by J.G. Thompson. If G is a p-group, then G has a CAP subgroup of class at most 2 ( and exponent p if p is odd or G is Abelian, of exponent 4 if p = 2 and G is non-Abelian) ( actually, Thompson imposed slightly different conditions). There are several ways to construct these: one way which is computationally efficient is to define CAP subgroups G_{n} of G as follows: G_{0} = G; if Φ(G_{n}) ≤ Z(G_{n}), set G_{∞} = Ω(G_{n}). Otherwise, set G_{n+1} = Φ(G_{n})C_{G_{n}}(Φ(G_{n})).

Notice that G_{n+1} is a proper characteristic subgroup of G_{n} which contains its centralizer in G_{n} in the case that Φ(G_{n}) ≰ Z(G_{n}), so that, in that case, G_{n+1} is CAP in G_{n}. On the other hand, if Φ(G_{n}) ≤ Z(G_{n}), then G_{n} has class at most 2, and G′_{n} has exponent at most p. Hence the procedure above returns a CAP subgroup G_{∞} of G of class at most 2 and exponent p or 4. Furthermore, G_{∞}/Φ(G_{∞}) is still a CAP section of G.

If desired, the procedure can be refined as follows ( but the resulting procedure is not as computationally efficient): suppose that we have reached the point that Φ(G_{n}) ≤ Z(G_{n}). If G_{n} has a characteristic Abelian subgroup which is not central, say A_{n}, we set G_{n+1} = C_{G_{n}}(A_{n}), which is still a CAP section of G. Hence we can produce a CAP subgroup G_{∞} which has exponent p or 4, has all characteristic Abelian subgroups central, and has central Frattini subgroup.

If a finite group G is contained in another finite group H and S = L/K is a section of G, then C_{H}(S) is defined to be {h ∈ H : [L, h] ≤ K} ( which is certainly a subgroup of N_{H}(L)).

DEFECT GROUPS, B-PARABOLICS, AND CAP SECTIONS

LEMMA: Let U be a subgroup of the finite p-group P, and let S be a CAP section of U. Then if C_{P} (S) ≰ U, we have C_{N_{P} (U)}(S) ≰ U. PROOF: Let C = C_{P} (S). Then U normalizes C, so that UC is a p-group. If C ≰ U, then UC > U, so that UN_{C}(U) = N_{UC}(U) > U. Hence N_{C}(U) ≰ U.

Now let B be a block of RG for a finite group G, and let D be a defect group for B. Let (D, b^{∗}) be a maximal B-subpair. Let U be a subgroup of D, and let (U, b) ≤ (D, b^{∗}) be a B-subpair such that (N_{D}(U), b_{1}) ≤ (D, b^{∗}) is a Sylow subpair of N_{G}(U), that is, a maximal B′-subpair for some Brauer correspondent of B for N_{G}(U). Suppose further that C_{D}(U) = Z(U).

Let S be a CAP section of U. Then

(C_{G}(S) ∩ N_{G}(U))/C_{G}(U) ≤ O_{p}(N_{G}(U)/C_{G}(U)).

If C_{D}(S) ≰ U, then C_{D}(S) ∩ N_{G}(U) > U. Now B′ covers a block of C_{G}(S) ∩ N_{G}(U) with defect group C_{D}(S) ∩ N_{D}(U) ≰ U. Since B′ covers a nilpotent block of C_{G}(U) ( with defect group Z(U)), we conclude that B′ covers a nilpotent block of U(C_{G}(S) ∩ N_{G}(U)) whose defect group strictly contains U.

By Theorem 3.2.2 of [9], we deduce that

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B′_{σ}) = 0

for each non-negative integer d. ( We first need to note that the alternating sum vanishes by the usual elementary argument if U ≠ O_{p}(N_{G}(U)). Similarly, the assumption that C_{D}(U) ≤ U above is no real loss of generality, since is is proved in [9] that, in the situations to be considered below, the alternating sum in question vanishes anyway if this is not the case).

Hence we have: THEOREM A : Let G be a minimal counterexample to DPC ( in the sense described in [7]), and B be a block of RG with defect group D for which some formula predicted by DPC fails. Let U > Z be a p-subgroup of G for which

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) ≠ 0

for some non-negative integer d. Then there is a conjugate, V say, of U, contained in D such that C_{D}(S) ≤ V for each CAP section, S of V. REMARK: Once more, this can be recast as a Theorem about groups in which DPC is valid in every section. The same remark applies to the two theorems below. In fact, the Theorem above can be strengthened somewhat: for, setting Ḡ = G/Z, etc., then if U is as above, and S̄ is a CAP section of Ū, then

(C_{G}(S̄) ∩ N_{G}(U))/C_{G}(U)

is still a p-group ( for if y ∈ N_{G}(U) is p-regular with ȳ ∈ CḠ(S̄), then [U, y] ≤ Z, so that [U, y, y] = 1).

Hence we have : THEOREM B : Let G be a minimal counterexample to DPC and B be a block of RG with defect group D for which some formula predicted by DPC fails. Let U > Z be a p-subgroup of G for which

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) ≠ 0

for some non-negative integer d. Then there is a conjugate, V say, of U, contained in D such that C_{D}(S̄) ≤ V for each CAP section, S̄, of V̄ , where Ḡ = G/Z, etc.

Theorem B admits yet further strengthening. For let G be as above, and let N be a non-central normal subgroup of G, containing Z. Let B cover a block B_{1} with defect group D_{0} of N ( we know from [9] that D_{0} > Z). Let us consider a p-subgroup U > Z of N, and the contribution to the relevant alternating sum in Theorem 1 of [9] from ( orbits of ) chains starting with p-subgroups V of G with V ∩ N = U. This is the same ( for a given d )

∑_{σ∈N (N,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ})

by the elementary arguments outlined in [9].

This last alternating sum is 0 by Theorem 2 of [9], unless we choose may choose a conjugate of U, say W, so that C_{D_{0}}(W) ≤ W. Hence we may suppose that C_{D_{0}}(U) = Z(U), and that N_{D_{0}}(U) is a defect group for some Brauer correspondent for RN_{N}(U) of a block of RN covered by B. By Theorem 3.2.2 of [9], for a given p-subgroup V of G with V ∩ N = U, the contribution from chains in N (N_{G}(V), V)/N_{G}(V) will be zero if some Brauer correspondent of B for N_{G}(V) covers a nilpotent block of a normal subgroup with defect group not contained in U. By the arguments above, there will be such a block unless we can choose a conjugate, W, of U so that C_{D}(S̄) ≤ W for each CAP section, S̄, of W̄ , where N̄ = N/Z, etc.. THEOREM C : Let G be a minimal counterexample to DPC and B be a block of RG with defect group D for which some formula predicted by DPC fails. Let N be a non-central normal subgroup of G. Let D_{0} = D ∩ N. Let U > Z be a p-subgroup of N for which

∑_{σ∈N (N,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) ≠ 0

for some non-negative integer d. Then there is a conjugate, W say, of U, contained in D_{0} such that C_{D_{0}}(S̄) ≤ W for each CAP section, S̄, of W̄ , where N̄ = N/Z, etc.

EXAMPLES¡ APPLICATIONS, PARTIAL RESULTS

Since Dade’s projective conjecture is proved in [7] for p-solvable groups, we may deduce:

THEOREM D : Let G be a finite p-solvable group such that O_{p}(G) = Z ≤ Z(G). Let B be a block of RG with defect group D. Let N be a non-central normal subgroup of G containing Z, and let D_{0} = D ∩ N. Let U > Z be a p-subgroup of N for which

∑_{σ∈N (N,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) ≠ 0

for some non-negative integer d. Then there is a conjugate, W say, of U, contained in D_{0} such that C_{D_{0}}(S̄) ≤ W for each CAP section, S̄, of W̄ , where N̄ = N/Z, etc.

Perhaps worth singling out is : COROLLARY E: Let G be a minimal counterexample to DPC and B be a block of RG with defect group for which some formula predicted by DPC fails. Let U > Z be a p-subgroup of G for which

∑_{σ∈N (G,U)/N_{G}(U)}(−1)^{|σ|}k_{d}(B_{σ}) ≠ 0

for some non-negative integer d. Then:

i) U/Z is not cyclic. ii) If Φ(U/Z) contains every elementary Abelian normal subgroup of U/Z, then U/Z contains every elementary Abelian subgroup of D/Z which it normalizes. PROOF: Set Ḡ = G/Z, etc.. If Ū is cyclic, then ND̄(Ū) ≤ CD̄(Ū/Φ(Ū)). By Theorem B, we must have D̄ = Ū. However, in that case, Dade has proved in [2] that G is not a counterexample to DPC. If Ω_{1}(Ū) ≤ Φ(Ū), then let Ā be any elementary Abelian subgroup of D̄ normalized by Ū. Supppose that Ā ≰ Ū. Since ŪĀ is a p-group strictly containing Ū, we have X̄ = NĀ(Ū) ≰ Ū. Then [Ū, X̄] ≤ Ū ∩ X̄ ≤ Φ(Ū) by hypothesis ( as [Ū, X̄] is an elementary Abelian normal subgroup of Ū). However, by Theorem B, we have CD̄(Ū/Φ(Ū)) ≤ Ū, a contradiction. REMARKS: The analogue of part i) of corollary E was already observed for Alperin’s conjecture in [5]. Notice that ( using P. Hall’s classification of p-groups of symplectic type), we may conclude from Corollary E that if Ū is (generalized) quaternion, then D̄ must be semi-dihedral or generalized quaternion ( and if Ū is not quaternion of order 8 with 3 dividing [NḠ(Ū) : CḠ(Ū)], then Ū = D̄).

Similar statements to Corollary E may be made about defect groups of covered blocks, but we don’t state them explicitly.

We may turn Theorem C around as follows: THEOREM F : Let G be a minimal counterexample to DPC and B be a block of RG with defect group D for which some formula predicted by DPC fails. Let N be a non-central normal subgroup of G containing Z and let D_{0} = D ∩ N. Suppose that there is no proper subgroup U of D_{0} such that C_{D_{0}}(S̄) ≤ U for each CAP section, S̄, of Ū, (where N̄ = N/Z, etc). Then there must be some non-negative integer d such that k_{d}(B) ≠ k_{d}(B′), where B′ is the Harris-Knörr correspondent ( see [4]) of B for N_{G}(D_{0}). REMARKS: Even when D/Z is Abelian and N = G, Theorem F says something interesting. For then it may be re-cast as saying that in that situation, DPC predicts that there should be a defect preserving bijection between irreducible characters in B and irreducible characters in the unique Brauer correspondent for N_{G}(D). When Z = 1, this is a predicted consequence of Broué’s conjecture for blocks with Abelian defect groups.

It follows from the results of [8] that if G is a minimal counterexample to DPC, but Z is non-trivial, and D/Z is Abelian, then there is a bijection between irreducible characters in B and irreducible characters in the unique Brauer correspondent of B for N_{G}(D), but it is not clear to us at present how to incorporate character defects to get the full strength of the desired refinement. Alternatively, if Broué’s conjecture were proved ( in the weaker perfect isometry version), then it would follow from [8] that there is a bijection between irreducible characters in B and irreducible characters in the unique Brauer correspondent for N_{G}(D) in the case that D/O_{p}(Z(G)) is Abelian.

We remark without proof that, by arguments similar to the above, we have : THEOREM G : Let G be a minimal counterexample to DPC and B be a block of RG with defect group D for which some formula predicted by DPC fails. Let N be a non-central normal subgroup of G containing Z, and let D_{0} = D ∩ N. Let B̄ be the unique block of Ḡ = G/Z which corresponds to B under the canonical epimorphism from RG onto RḠ. Then if B̄ covers blocks of N̄ which have TI defect groups, there must be some non-negative integer d such that k_{d}(B) ≠ k_{d}(B′), where B′ is the Harris-Knörr correspondent of B for N_{G}(D_{0}).

References

[1] Alperin,J.L., Weights for Finite Groups, Proceedings of Symposia in Pure Mathematics 47, ( AMS, Providence,1987),369-379.

[2] Dade, E.C., Counting Characters in Blocks with cyclicdefect group, Journal of Algebra, 186,(1996), 934-969.

[3] Gorenstein,D.,Finite Groups,Harper and Row, New York, (1968).

[4] Harris, M;Knörr, R., Brauer correspondence for covering blocks of finite groups, Comm. Alg. 13,(1985),1213-1218.

[5] Knörr, R.; Robinson, G.R., Some remarks on a conjecture of Alperin, J. London Math. Soc., (2),39,(1989), 48-60.

[6] Külshammer, B.; Puig, L., Extensions of Nilpotent Blocks, Invent. Math,102,1,(1990), 17-71.

[7] Robinson, G.R., Dade’s Projective Conjecture for p-Solvable Groups, J. Algebra, 229,(2000),234-248.

[8] Robinson, G.R., On a projective generalization of Alperin’s conjecture,Algebras and Representation Theory, 1,(1998),129-134.

[9] Robinson, G.R., Cancellation Theorems related to conjectures of Alperin and Dade, submitted.