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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Let G be a finite p-group of order pn and M(G) be the Schur multiplier of G. In 1956, Green proved that |M(G)| = p(n(n-1)/2)-t(G), where t(G) ≥ 0. Berkovich (1991), Zhou (1994) and Ellis (1999) have determined the structure of G, when t(G) = 0, 1, 2 and 3, respectively. In 2007, Salemkar et al. classified the structure of G for t(G) = 4 under one condition. In this paper, we characterize all finite abelian p-groups for t(G) ≥ 0 and non-abelian p-groups, for t(G) = 4 and 5.

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