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Let G be a group and a be an automorphism of G.  In 2016, Ganjali and Erfanian introduced the notion of a normal subgroup of G which is relative to a, called an a-normal subgroup.  Suppose that N is an a-normal subgroup of G.  In this research, we show that N is t-normal in G for an automorphism t in Aut(G) if and only if a(g)t(g^-1) is in N for every g in G.

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