亚环的极大环子群的共轭类𝑝-groups

Pub Date : 2022-10-15 DOI:10.1515/jgth-2022-0103

M. Bianchi, R. Camina, M. Lewis

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引用次数: 0

摘要

摘要:本文设η (G) \eta (G)为有限群𝐺的极大循环子群的共轭类的个数。我们计算所有元环𝑝-groups的η∑(G) \eta (G)。我们证明了如果𝐺是p n p^{n}阶的元环𝑝-group，它不是二面体、广义四元数或半二面体，则η∑(G)≥n-2 \eta (G) \geq n-2，并确定了等式在什么时候成立。

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Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups

Abstract In this paper, we set η ⁢ ( G ) \eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G ) \eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2 \eta(G)\geq n-2 , and we determine when equality holds.

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