# A question regarding conjugacy classes of central involutions.

An involution $$a$$ of a group $$G$$ is called central if there exists a sylow $$2$$-subgroup $$H$$ of $$G$$ such that $$a \in C_G(H)$$.

Clearly if an involution is central then its every conjugate is also central.

1. If $$C$$ is a conjugacy class of involutions and $$a$$, $$b$$ are two distinct members of $$C$$ that commute with each other then is $$ab$$ also a member of $$C$$?

2. Can we classify all finite groups (or finite simple groups) for which the number of conjugacy classes of central involutions is $$1$$?

Let $$G = \langle (3, 4), (1, 3)(2, 4) \rangle \cong D_8$$. Set $$a = (1, 3)(2, 4)$$ and $$b = (1, 4)(2, 3)$$. Then $$a^2 = b^2 = \operatorname{id}$$, $$ab = ba$$ and $$C = \{ a, b \}$$ is a conjugacy class of $$G$$, but $$ab = (1, 2)(3, 4) \not\in C$$.