Title: Semi-extraspecial groups with an abelian subgroup of maximal possible order,
Abstract: 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 $p^{3n}$ that contain an abelian
subgroup of order $p^{2n}$. In the literature, it has been proved that every
ultraspecial group $G$ 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.