Some periodic groups admitting a finite regular automorphism of even order (Q2011318)

The authors are interested in studying infinite groups \(G\) with regular automorphisms \(\phi\) of finite order \(2p\) where \(p\) is an odd prime, the finite case having been studied extensively some 50 years ago, mainly by B. Fischer. The main interest here is in the case where \(G\) is locally finite, but the authors can regularly relax this assumption to periodic groups satisfying hypotheses such as every element of \(G\) lies in a finite \(\phi\)-invariant subgroup and the centralizers of involutions are locally finite. The vast bulk of this paper concerns groups \(H\) with elements \(i\) and \(a\), where \(i\) has order 2 and \(a\) has order \(p>2\), such that for every \(g\in H\backslash C_H(i)\) the subgroup \(\langle i,a^g\rangle\) is a finite Frobenius group with the non-normal factor of even order and containing \(a g\). The connection with \(G\) and \(\phi\) above is that \(H\) might be the split extension of \(G\) by \(\langle\phi\rangle\).