Let G be an arbitrary-group, where-groups are groups with number of centralizers & is any finite number. In this article, we have proved that the group of inner automorphisms of G is isomorphic to some other groups depending upon. Moreover if for some group , the group of inner automorphisms () has order 6 or 9 then will be 5-group and if for some group , the group of inner automorphisms () has order 4 then will be 4-group & conversely.
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