Semi-extraspecial groups with an abelian subgroup of maximal possible order


Let pp be a prime. A pp-group GG is defined to be semi-extraspecial if for every maximal subgroup NN in Z(G)Z(G) the quotient G/NG/N is a an extraspecial group. In addition, we say that GG is ultraspecial if GG is semi-extraspecial and G:G=G2|G:G'| = |G'|^2. In this paper, we prove that every pp-group of nilpotence class 22 is isomorphic to a subgroup of some ultraspecial group. Given a prime pp and a positive integer nn, we provide a framework to construct of all the ultraspecial groups order p3np^{3n} that contain an abelian subgroup of order p2np^{2n}. In the literature, it has been proved that every ultraspecial group GG order p3np^{3n} with at least two abelian subgroups of order p2np^{2n} can be associated to a semifield. We provide a generalization of semifield, and then we show that every semi-extraspecial group GG that is the product of two abelian subgroups can be associated with this generalization of semifield

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