20.93 (2022)

A group $G$ is called a Shunkov group if for any finite subgroup $H \leqslant G$ any two conjugate elements of prime order in $N_G(H)/H$ generate a finite subgroup. Are the following well-known results of the theory of finite groups true in the class of (periodic) Shunkov groups?
$\qquad$ a) The Baer–Suzuki theorem (see 11.11).
$\qquad$ b) The Burnside–Brauer–Suzuki theorem on the existence of a normal section of order 2 in a group with a non-trivial Sylow 2-subgroup containing only one involution (see 4.75).
$\qquad$ c) Glauberman’s Z$^*$-theorem (see 10.62 and 11.13).