On p-groups with a maximal elementary abelian normal subgroup of rank k

There are several results in the literature concerning p-groups G with a maximal elementary abelian normal subgroup of rank k due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators etc. of a 2-group G in terms of k, which were previously known only for p > 2. We also prove a theorem that is new even for odd primes. Namely, we show that if G has a maximal elementary abelian normal subgroup of rank k, then for any abelian subgroup A the Frattini subgroup Phi(A) can be generated by 2k elements (3k when p = 2). The proof of this rests upon the following result of independent interest: If V is an n-dimensional vector space, then any commutative subalgebra of End(V) contains a zero algebra of codimension at most n. (c) 2024 Elsevier Inc. All rights reserved.