10.27 (1986)

a) Let $t$ be an involution of a finite group $G$ and suppose that the set $D = t^G \cup \{t^x t^y \mid x, y \in G, \ \lvert t^x t^y \rvert = 2\}$ does not intersect $O_2(G)$. Prove that if $t \in O_2(C(d))$ for any involution $d$ from $C_D(t)$, then $D = t^G$ (and in this case the structure of the group $\langle D \rangle$ is known).
b) A significantly more general question. Let $t$ be an involution of a finite group $G$ and suppose that $t \in Z^*(N(X))$ for every non-trivial subgroup $X$ of odd order which is normalized, but not centralized, by $t$. What is the group $\langle t^G \rangle$?