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

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On the rings generated by the inner automorphisms of finite groups

  • 期刊论文

  • 发表日期：

    2025年7月16日

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  • Wen-Fong Ke,
  • Chun-Wei Ting

摘要

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 pn (with p being a prime and n = 3 or 4), as well as for orders 35 and 36, we determine the ring structure of I(G).