Abstract In this paper, we set ? ? ( G stretchy="false">) \eta(G) to be the number of conjugacy classes maximal cyclic subgroups a finite group ????. We compute for all metacyclic ????-groups. show that if ???? is ????-group order <m:msup...

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$ possesses $3$-class $C_{\mathrm{k},3}$ type $(9,3)$, capitulation $3$-ideal classes in its unramified cyclic cubic extensions determine...

Let p be an odd prime. We construct a non-abelian extension Γ of S1 by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on...

Let L be a restricted Lie algebra with the restricted enveloping algebra u(L) over a perfect field of positive characteristic p. The restricted isomorphism problem asks what invariants of L are determined by u(L). This problem is the analogue of the modular isomorphism problem for finite p-groups. Bagiński and Sandling have given a positive answer to the modular isomorphism problem for metacycl...

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on vector space V, is not metacyclic. Then always has regular orbit V except for few “small” cases. We completely classify these cases in this paper.