arXiv subjects
Publications and source records attributed to Humberto Bautista Serrano.
We obtain a complete description of the integer group determinants for SmallGroup(16,8), the semidihedral group of order 16. While this paper was in preparation, a complete descriptions for this group was independently obtained by Yuka Yamaguchi and Naoya Yamaguchi in [21] (and the other remaining group of order 16 in [20]). We offer our version here for anyone interested in a slightly different approach.
We obtain a complete description of the integer group determinants for SmallGroup(16,13), the central product of the dihedral group of order eight and cyclic group of order four. These values are the same as the integer group determinants for SmallGroup(16,11), the direct product of the dihedral group of order eight and cyclic group of order two. It was not previously known that the integer group determinants do not determine the group.
The covering number of a nontrivial finite group $G$, denoted $σ(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $ι(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $ι(G)$ when $G$ is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.