Semi-Extraspecial Groups with an Abelian Subgroup of Maximal Possible Order

Abstract

Let p be a prime. A finite 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 finite p-group of nilpotence class 2 and exponent p is isomorphic to a subgroup of some ultraspecial group. Given a prime p and a positive integer n, we provide a framework for the construction of all the ultraspecial groups of order p^{3n} that contain an abelian subgroup of order p^{2n}. In the literature, it has been proved that every ultraspecial group G of order p^{3n} with at least two abelian subgroups of order p^{2n} 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.

Keywords

• semi-extraspecial group
• ultraspecial group
• semifield