On \(G\)-character tables for normal subgroups. (English) Zbl 1517.20011
Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\). Since \(N\) is a union of \(G\)-conjugacy classes, it is natural to wonder whether those columns of the character table of \(G\) provide structural information of \(N\) (for an overview of this topic see [A. Beltran et al., Int. J. Group Theory 7, No. 1, 23–36 (2018; Zbl 1446.20047)]).
From a result of R. Brauer [Lect. Modern Math. 1, 133–175 (1963; Zbl 0124.26504)], it can be derived that the character table of \(G\) contains square submatrices which are induced by the \(G\)-conjugacy classes of elements in \(N\) and the \(G\)-orbits of irreducible characters of \(N\). In the paper under review, the authors provide an alternative approach to this fact through the structure of the group algebra. They also show that such matrices are non-singular and become a useful tool to obtain information of \(N\) from the character table of \(G\).
From a result of R. Brauer [Lect. Modern Math. 1, 133–175 (1963; Zbl 0124.26504)], it can be derived that the character table of \(G\) contains square submatrices which are induced by the \(G\)-conjugacy classes of elements in \(N\) and the \(G\)-orbits of irreducible characters of \(N\). In the paper under review, the authors provide an alternative approach to this fact through the structure of the group algebra. They also show that such matrices are non-singular and become a useful tool to obtain information of \(N\) from the character table of \(G\).
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20C15 | Ordinary representations and characters |
| 20E45 | Conjugacy classes for groups |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
Software:
GAPReferences:
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| [12] | The GAP Group: GAP - Groups, Algorithms, and Programming, Version 4.11.1, 2021. https://www.gap-system.org. |
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