Acentralizers of some Finite Groups

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Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.