Abstract. For a finite group G, let F 2 (G) be the number of factorizations G = AB of the group G, where A and B are subgroups of G. We compute F 2 (G) for certain classes of groups, including cyclic groups ޚ n , elementary abelian p-groups ޚ n p , dihedral groups D 2n , generalised quaternion groups Q 4n , quasi-dihedral 2-groups QD 2 n (n ≥ 4), modular p-groups M p n , projective general linear groups PGL(2, p n ) and projective special linear groups PSL(2, p n ).2000 Mathematics Subject Classification. Primary 20D40, secondary 20P05.
In 1975, Sherman introduced the probability of an automorphism of a finite group, which fixes an arbitrary element of the group. In this paper we introduce a new probability concept, namely the probability of an automorphism of a given finite group such that it fixes a subgroup element of the group. Among other results, we construct some upper and lower bounds for both probabilities.
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