In search of nonsolvable groups of central type

Liebler, Robert A.; Yellen, Jay

doi:10.2140/pjm.1979.82.485

Cited by 20 publications

(21 citation statements)

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“…There is a considerable literature[4,5,7,8,12,14] on finite groups G with a faithful irreducible character of degree √ |G/Z(G)|, which is the largest possible degree. It is known that such groups are soluble[7,12], and that every finite soluble group embeds in G/Z(G) for some such group G[5].…”

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confidence: 99%

Blocks inequivalent to their Frobenius twists

Kessar

2007

Journal of Algebra

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Let k be an algebraically closed field of characteristic p. We give a general method for producing examples of blocks B of finite group algebras that are not Morita equivalent as k-algebras to the Frobenius twist B (p) . Our method produces non-nilpotent blocks having one simple module and elementary abelian defect group. These also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive, but no derived equivalence between the blocks.We do not know of any examples of blocks B that are not Morita equivalent to the second Frobenius twist B (p 2 ) .

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confidence: 99%

Blocks inequivalent to their Frobenius twists

Kessar

2007

Journal of Algebra

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“…In other words, H admits an irreducible projective representation of dimension n = √ |H |. If char F = 0, this implies that H is solvable [34].…”

Section: Definition 52 [12] a G-grading Onmentioning

confidence: 99%

Gradings on Finite-Dimensional Simple Lie Algebras

Kochetov

2008

Acta Appl Math

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“…Let C = CG(K), and V = hr(K). Let R = R/C be a chief factor of G. Then R acts faithfully on V and C v {k) is trivial, so that R has order coprime to p. Thus we may use the arguments in the proofs of Lemmas 2.4 and 2.5 of [10] to show that some S e Proj (G, a) is G-invariant. Let v be a nontrivial element of V, then I G (Sv) = IQ(V) = Ia(v~l) = IG(SV~1).…”

Section: Let G Be a Group Of Odd Order Then Either G Is Of A-centralmentioning

confidence: 99%

On projective characters of the same degree

Higgs

1998

Glasgow Math. J.

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The aim of this short paper is to provide some positive evidence about the following conjecture, of which the result mentioned above is just a special case. CONJECTURE. Let G be a group and a be a cocycle of G. Then either G is of a-central type or Proj(G, a) contains at least two elements of the same degree.The reader will discover that groups of a-central type play an important part in our investigation of the conjecture, which we are able to verify in a number of cases; most notably when G is supersoluble or has odd order.

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In search of nonsolvable groups of central type

Cited by 20 publications

References 11 publications

Blocks inequivalent to their Frobenius twists

Blocks inequivalent to their Frobenius twists

Gradings on Finite-Dimensional Simple Lie Algebras

On projective characters of the same degree

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