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Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of [5], which avoids the use of the classification of finite simple groups.

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p-groups of order {p^{5}}. As an application, we determine the capability of groups of order {p^{5}}.

For a group G and a subset X of G, the commuting graph of X, denoted by Γ(G, X) is the graph whose vertex set is X and any two vertices u and v in X are adjacent if and only if they commute in G. In this article, certain properties of the commuting graph of generalized dihedral groups have been studied.

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