Rationality of blocks of quasi-simple finite groups

Farrell, Niamh; Kessar, Radha

doi:10.48550/arxiv.1805.02015

Cited by 1 publication

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“…Proof. This follows from Theorem 1.2 and [9], in which it is proved that sf O (B) ≤ 4|D| 2 ! for all blocks B of quasisimple finite groups with defect groups D. Note that the setting of [9] Proof.…”

Section: Reductions For Donovan's Conjecturementioning

confidence: 71%

“…This follows from Theorem 1.2 and [9], in which it is proved that sf O (B) ≤ 4|D| 2 ! for all blocks B of quasisimple finite groups with defect groups D. Note that the setting of [9] Proof. This follows immediately from Corollary 4.5 and [6, 9.2], in which it is proved that the Cartan invariants of 2-blocks with abelian defect groups are bounded in terms of the defect.…”

Section: Reductions For Donovan's Conjecturementioning

confidence: 71%

“…We therefore have two canonical choices for O in the statement of Donovan's conjecture: the ring of Witt vectors for Fp and the same with a primitive |P | th root of unity attached. The results of this paper hold over either choice (see Remark 4.7 for the latter case), but in light of the results of [9] the former seems the best setting for Donovan's conjecture.…”

Section: Introductionmentioning

confidence: 67%

“…For blocks defined over an algebraically closed field there has been some success in proving the conjecture for certain p-groups, for example it is known for abelian 2-groups by [5] and for abelian defect groups in arbitrary characteristic it reduces to bounding the Cartan invariants of blocks of quasisimple groups by [9] and [5]. In addition blocks with defect groups isomorphic to dihedral or semidihedral 2-groups were classified in [8].…”

Section: Introductionmentioning

confidence: 99%

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Donovan’s conjecture, blocks with abelian defect groups and discrete valuation rings

2019

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We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring O. Consequences are that Donovan's conjecture holds for O-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for O-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect.A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for O-blocks is a consequence of conjectures predicting bounds on the O-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field. 1 studying groups generated by the defect groups, was only known over a field. The second is that the reduction in [12] of Donovan's conjecture into two distinct conjectures was also only known over a field. The first problem was overcome by the second author in [7], and we resolve the second here, allowing us to reduce Donovan's conjecture for O-blocks with abelian defect groups to bounding, for quasisimple groups, the Cartan invariants and strong Frobenius number as defined in [4]. The results of [9] show that the strong Frobenius numbers of quasisimple groups are bounded in terms of the defect group, so Donovan's conjecture for abelian defect groups in fact reduces to bounding Cartan invariants of blocks of quasisimple groups. Such bounds are known to hold for 2-blocks with abelian defect groups.Our main result is as follows:Theorem 1.2. Let d be a non-negative integer. If there are functions s, c : N → N such that for all O-blocks B of quasisimple groups with abelian defect groups of order p d ′ dividing p d , sf O (B) ≤ s(d ′ ) and all Cartan invariants are at most c(d ′ ), then Donovan's conjecture holds for O-blocks with abelian defect groups of order p d . A straightforward consequence is that: Corollary 1.3. Donovan's conjecture (over O) holds for blocks with abelian defect groups if and only if it holds for blocks of quasisimple groups with abelian defect groups. By work of Farrell and Kessar in [9], we get a much more powerful consequence: Corollary 1.4. Let d be a non-negative integer. If there is a function c : N → N such that for all O-blocks B of quasisimple groups with abelian defect groups of order p d ′ dividing p d the Cartan invariants are at most c(d ′ ), then Donovan's conjecture holds for O-blocks with abelian defect groups of order p d .Hence we have shown that for abelian p-groups Conjecture 1.1 is equivalent to (the restriction to quasisimple groups of) the following apparently much weaker conjecture, which arose from a question of Brauer:Conjecture 1.5 (Weak Donovan). Let P be a finite p-group. Then there is c(P ) ∈ N such that if G is a finite group and B is a block of kG with defect groups isomorphic to P , then the entries of the Cartan matrix of B are at most c(P ).Remark 1.6. It actuall...

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Section: Reductions For Donovan's Conjecturementioning

confidence: 71%

Section: Reductions For Donovan's Conjecturementioning

confidence: 71%

Section: Introductionmentioning

confidence: 67%