Inducing π\pi-partial characters with a given vertex

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup of a subgroup $V$. Suppose \phi \in \ibr G. If either $|G|$ is odd or $p = 2$, we prove that the number of Brauer characters of $H$ inducing $\phi$ with vertex $Q$ is at most |\norm GQ: \norm VQ|

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

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