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Conjugacy classes of maximal cyclic subgroups. (arXiv:2201.05637v1 [math.GR]) http://arxiv.org/abs/2201.05637

Conjugacy classes of maximal cyclic subgroups

In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $η(G/N) = η (G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$.

arxiv.org
January 19, 2022 at 3:10 AM · · feed2toot · 0 · 0 · 0
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