On a question by V. P. Shunkov (Q1288082)

The author gives a complete answer to a question by \textit{V. P. Shunkov} [Sib. Mat. Zh. 34, No. 2, 208-219 (1993; Zbl 0835.20051)]. Let \(G\) be a group with involutions and let \(i\) be some involution of \(G\) satisfying the following conditions: (i) every subgroup \(\langle i,i^g\rangle\) generated by \(i\) and \(i^g\), \(g\in G\), is finite; (ii) every Sylow 2-subgroup of \(G\) is cyclic or a generalized quaternion group; (iii) the centralizer \(C_G(i)\) is infinite and has finite periodic part; (iv) the normalizer of every nontrivial \(i\)-invariant locally finite subgroup of \(G\) either is contained in \(C_G(i)\) or has periodic part that is a Frobenius group with abelian kernel and finite complement of even order; (v) the inequality \(C_G(i)\neq G\) holds and, for every element \(c\in G\setminus C_G(i)\) satisfying \(c^i=c^{-1}\), there exists an element \(s_c\) such that the group \(\langle c,c^{s_c}\rangle\) is infinite. If these conditions hold, the group \(G\) is called a \(T_0\)-group. Another class of groups under consideration is the class of \(\Phi_0\)-groups. These are groups with involution \(i\) such that the following conditions are satisfied: (i) the same as above; (ii) the centralizer \(C_G(i)\) is infinite and has finite periodic part; (iii) \(C_G (i)\neq G\) and \(C_G (i)\) is not contained in other subgroups of \(G\) with periodic part; (iv) if \(K\) is a finite subgroup of \(G\) not in \(C_G(i)\) and \(V=K\cap C_G(i)\neq 1\) then \(K\) is a Frobenius group with complement \(V\). The question by Shunkov is as follows: Is it true that every \(\Phi_0\)-group is also a \(T_0\)-group? The author proves that, for every \(\Phi_0\)-group, all the conditions except (iv) of the definition of \(T_0\)-groups are satisfied. He presents an example of a \(\Phi_0\)-group that is not a \(T_0\)-group.