%0 Journal Article %T Some Characterizations of a Normal Subgroup of a Group %A Vipul Kakkar %A R. P. Shukla %J Mathematics %D 2012 %I arXiv %X 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 %U http://arxiv.org/abs/1202.5626v2