The groups whose cyclic subgroups are either ascendant or almost self-normalizing (Q2831610)

The subgroup \(H\) of \(G\) is ascendant in \(G\) if the chain of normalizers beginning with \(H\) ends (possibly transfinitely) with \(G\). \(H\) is almost selfnormalizing if \(N_G(H)/H\) is finite. The authors characterize infinite locally finite groups of the title. In this connection, the Gruenberg radical \(\mathrm{Gru}(G)\), the subgroup generated by all cyclic ascending subgroups of the group \(G\), is important. The groups \(G\) under consideration have the following structure:{\parindent=0.7cm \begin{itemize} \item[(a)] \(G/\mathrm{Gru}(G)\) is finite and \(\mathrm{Gru}(G)\) is nilpotent-by-finite, \item [(b)] if \(g \not\in \mathrm{Gru}(G)\) then \(C_G(g)\) is finite, \item [(c)] if \(\sigma \) is the set of primes such that \(\sigma = \pi (G/\mathrm{Gru}(G))\), then \(G\) is the split extension of a normal Hall \(\sigma '\)-subgroup \(Q\) by a Hall \(\sigma \)-subgroup \(R\) of \(G\), further \(R\) is a Chernikov group and \(\mathrm{Gru}(G) = C_R(Q) \times Q\). NEWLINENEWLINE\end{itemize}} Further information is given for the quotient group \(F = G/\mathrm{Gru}(G)\): If \(Q\) is infinite, all Sylow subgroups of \(F\) are cyclic or generalized quaternion and subgroups of order \(pq\) are cyclic. If \(Q\) is finite and the divisible part \(D\) of \(G\) is a \(p\)-group, all Sylow subgroups except possibly the Sylow \(p\)-subgroups are cyclic or generalized quaternion, on the other hand, if \(Q\) is finite and \(D\) is not a \(p\)-group, all Sylow subgroups are cyclic or generalized quaternion (Theorem A and corollaries). Modifications are given for the substitution of ``ascendant'' by ``subnormal'' and ``normal''.