Published by De Gruyter July 22, 2022

The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups

William Murphy

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2021-0176

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Abstract

Let 𝐺 be one of the sporadic simple Mathieu groups M 11 , M 12 , M 22 , M 23 or M 24 , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k ⁢ G . In particular, we calculate the dimension of HH 1 ⁢ ( B ) for the 𝑝-blocks 𝐵 of k ⁢ G , and in almost all cases, we determine whether HH 1 ⁢ ( B ) is a solvable Lie algebra.

A The dimension of the first Hochschild cohomology group of a finite group algebra

Using the following result, we are able to construct a simple GAP code to find the dimensions of the degree one Hochschild cohomology of a group algebra, coming from the centraliser decomposition.

Lemma A.1

The 𝑘-vector space Hom ⁢ ( G , k ) is non-zero if and only if 𝐺 has a quotient isomorphic to a non-trivial 𝑝-group. In particular,

dim k ⁡ ( Hom ⁢ ( G , k ) ) = dim k ⁡ ( Hom ⁢ ( R / Φ ⁢ ( R ) , k ) ) ,

where R = G / O p ⁢ ( G ) .

Once a group 𝐺 is defined in GAP, the command div below gives the set of prime divisors of | G | , π ⁢ ( G ) . The functions following div then take as input either 𝐺, or 𝐺 and a prime p ∈ π ⁢ ( G ) , and output the following lists respectively:

the centralisers of a complete set of conjugacy class representatives of 𝐺,
for each conjugacy class representative x ∈ G / ∼ , the groups
C G ⁢ ( x ) / O p ⁢ ( C G ⁢ ( x ) ) = R ,
the elementary abelian 𝑝-groups R / Φ ⁢ ( R ) ,
the rank of R / Φ ⁢ ( R ) .

The final function then produces a list of dimensions dim k ⁡ ( HH 1 ⁢ ( k ⁢ G ) ) , with one entry for each p ∈ π ⁢ ( G ) .

div:=PrimeDivisors(Size(G));

ListFpCentralisers:=function(X); > return List(ConjugacyClasses(X),i-> Image(IsomorphismFpGroup(Centralizer(X,Representative(i))))); > end;;

ListMaxPQuot:=function(X,p); > return List(ListFpCentralisers(X),i -> Image(EpimorphismPGroup(i,p))); > end;;

ListElAb:=function(X,p); > return List(ListMaxPQuot(X,p), i -> i/FrattiniSubgroup(i)); > end;;

ListPRank:=function(X,p); > return List(ListElAb(X,p), i -> RankPGroup(i)); > end;;

dimHH1:=function(X); > return List(div,i->Sum(ListPRank(X,i))); > end;

Acknowledgements

I am very grateful to the referee for their helpful and constructive comments, in particular with regards to the structure of this paper. I would also like to thank Xin Huang for his helpful comments on an earlier version of this paper, and Markus Linckelmann for his support and for many interesting conversations.

Communicated by: Olivier Dudas

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Received: 2021-10-26

Revised: 2022-03-10

Published Online: 2022-07-22

Published in Print: 2023-01-01