2017/10/27 by Mark L. Lewis, Lewis, Mark L.
Computer Science · Engineering · Mathematics · #20D15 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems #math.GR #msc:20D15
paper · pdf · doi:10.48550/arxiv.1710.10299
arxiv created 2017/10/27 · openalex publication_date 2017/10/27 · arxiv updated 2017/10/31 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28
Let p be a prime. A p-group G is defined to be semi-extraspecial if for every maximal subgroup N in Z(G) the quotient G/N is a an extraspecial group. In addition, we say that G is ultraspecial if G is semi-extraspecial and |G:G'| = |G'|2. In this paper, we prove that every p-group of nilpotence class 2 is isomorphic to a subgroup of some ultraspecial group. Given a prime p and a positive integer n, we provide a framework to construct of all the ultraspecial groups order p3n that contain an abelian subgroup of order p2n. In the literature, it has been proved that every ultraspecial group G order p3n with at least two abelian subgroups of order p2n can be associated to a semifield. We provide a generalization of semifield, and then we show that every semi-extraspecial group G that is the product of two abelian subgroups can be associated with this generalization of semifield.