2021/01/14 by J. E. Williams, Williams, James
Computer Science · Engineering · Mathematics · #20D15 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2101.05720
openalex publication_date 2021/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we investigate the structure of finite p-groups with the property that every subgroup of index pi is powerful for some i. For odd primes p, we show that under certain conditions these groups must be potent. Then, motivated by a question of Mann, we investigate in detail the case when all maximal subgroups are powerful. We show that for odd p any finite p-group G with all maximal subgroups powerful has a regular power structure - with precisely one exceptional case which is a 3-group of maximal class and order 81. To show this counterexample is unique we use a computational approach. We briefly discuss the case p=2 and some generalisations.