2023/05/03 by Zoltán Halasi, Halasi, Zoltán, Károly Podoski +5
Mathematics · Social Sciences · #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Global Educational Reforms and Inequalities #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2305.02037
openalex publication_date 2023/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 Φ(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.