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4.23 (1973)

Solved

Let $G$ be a finite simple group, $\tau$ some element of prime order, and $\alpha$ an automorphism of $G$ whose order is coprime to $|G|$. Suppose $\alpha$ centralizes $C_G(\tau)$. Is $\alpha = 1$?

Contributor: G. Glauberman

Progress

Not always; for example, $G = \text{Sz}(8)$, $|\tau| = 5$, $|\alpha| = 3$ (N. D. Podufalov, Letter of September, 3, 1975).

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