Abstract
Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such that $|B_i| \gt 1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n = 1,2,3$. Finally, we show that for a nonabelian group $G$ having a strict $n$-split decomposition, the index $|G:A|$ is bounded by some function of $n$.
Citation
Mark Lanning Lewis. Daria V. Lytkina. Viktor Danilovich Mazurov. Ali Reza Moghaddamfar. "Splitting via Noncommutativity." Taiwanese J. Math. 22 (5) 1051 - 1082, October, 2018. https://doi.org/10.11650/tjm/180202
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