Twisted cyclic groups

  • Neil Flowers

    Youngstown State University, USA
  • Thomas P. Wakefield

    Youngstown State University, USA
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A finite group GG is said to be twisted cyclic if there exist ϕAut(G)\phi \in \operatorname{Aut}(G) and xGx \in G such that G={(xi)ϕj ⁣:i,jZ}G=\{(x^i)\phi^j\colon i,j \in \mathbb{Z}\}. In this note, we classify all groups satisfying this property and determine that, if a finite group GG is twisted cyclic, then GG is isomorphic to Zpn\mathbb{Z}_{p^n}, Zp×Zp××Zp\mathbb{Z}_p\times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p, Q8Q_8, Zpn×Zpn\mathbb{Z}_{p^n}\times \mathbb{Z}_{p^n} or direct products of these groups for some prime pp and some nZ+n\in \mathbb{Z}^{+}.

Cite this article

Neil Flowers, Thomas P. Wakefield, Twisted cyclic groups. Rend. Sem. Mat. Univ. Padova 141 (2019), pp. 143–154

DOI 10.4171/RSMUP/18