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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 ever 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 $1$?

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