Article
| Title: | On the rings generated by the inner automorphisms of finite groups (English) |
| Author: | Ke, Wen-Fong |
| Author: | Ting, Chun-Wei |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 3 |
| Year: | 2025 |
| Pages: | 1029-1048 |
| Summary lang: | English |
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| Category: | math |
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| 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 $I(G)$. (English) |
| Keyword: | I-group |
| Keyword: | ring of inner automorphism of a group |
| Keyword: | nilpotent group |
| Keyword: | inner automorphism nearring |
| MSC: | 16S60 |
| MSC: | 16Y30 |
| MSC: | 20D15 |
| MSC: | 20F45 |
| DOI: | 10.21136/CMJ.2025.0070-25 |
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| Date available: | 2025-09-19T12:09:17Z |
| Last updated: | 2025-09-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153065 |
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| Reference: | [6] Levi, F., Waerden, B. L. van der: Über eine besondere Klasse von Gruppen.Abh. Math. Semin. Hamb. Univ. 9 (1933), 154-158 German. Zbl 0005.38507, MR 3069591, 10.1007/BF02940639 |
| Reference: | [7] Meldrum, J. D. P.: Near-Rings and their Links with Groups.Research Notes in Mathematics 134. Pitman, Boston (1985). Zbl 0658.16029, MR 0854275 |
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