12.101 (1992)

We call a group $G$ containing an involution $i$ a $T_0$-group if
$\qquad$ 1) the order of the product of any two involutions conjugate to $i$ is finite;
$\qquad$ 2) all 2-subgroups of $G$ are either cyclic or generalized quaternion;
$\qquad$ 3) the centralizer $C$ of the involution $i$ in $G$ is infinite, distinct from $G$, and has finite periodic part;
$\qquad$ 4) the normalizer of any non-trivial $i$-invariant finite subgroup in $G$ either is contained in $C$ or has periodic part which is a Frobenius group (see 6.55) with abelian kernel and finite complement of even order;
$\qquad$ 5) for every element $c$ not contained in $C$ for which $ci$ is an involution there is an element $s$ of $C$ such that $\langle c, c^s \rangle$ is an infinite subgroup.

Does there exist a simple $T_0$-group?