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Keywords:
I-group; ring of inner automorphism of a group; nilpotent group; inner automorphism nearring
I-group; ring of inner automorphism of a group; nilpotent group; inner automorphism nearring
Summary:
For a finite group $G$, let $I(G)$ denote the set of all finite sums of inner automorphisms of $G$. When $I(G)$ forms a ring, $G$ is referred to as an I-group. It is known that if $G$ is an I-group, then it is nilpotent of class at most 3, and that $I(G)$ is a commutative ring if and only if $G$ is nilpotent of class at most 2. We characterize the ring $I(G)$ for an I-group $G$. Additionally, for cases where $I(G)$ is a commutative ring and $G$ is of order $p^{n}$ (with $p$ being a prime and $n=3$ or 4), as well as for orders $3^{5}$ and $3^{6}$, we determine the ring structure of $I(G)$.
References:
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