Powers of conjugacy classes in a finite group. (English) Zbl 1471.20022
The authors consider how the decomposition of a product of conjugacy classes of a finite group \(G\) influence the structure of \(G\).
Particular emphasis is placed on the case when the power of a single conjugacy class is a union of one or two conjugacy classes, often with the aim of demonstrating solvability of \(G\). A typical example of the results is provided by Theorem \(B\): If \(K\) is a conjugacy class of \(G\) such that \(K^{n} = \{1_{G} \} \cup D\) for a single conjugacy class \(D\) and an integer \(n \geq 2\), then the elements of \(K\) generate a solvable (normal) subgroup of \(G\). The theory of complex characters is an essential tool in the proofs.
Particular emphasis is placed on the case when the power of a single conjugacy class is a union of one or two conjugacy classes, often with the aim of demonstrating solvability of \(G\). A typical example of the results is provided by Theorem \(B\): If \(K\) is a conjugacy class of \(G\) such that \(K^{n} = \{1_{G} \} \cup D\) for a single conjugacy class \(D\) and an integer \(n \geq 2\), then the elements of \(K\) generate a solvable (normal) subgroup of \(G\). The theory of complex characters is an essential tool in the proofs.
Reviewer: Geoffrey Robinson (Aberdeen)
MSC:
| 20E45 | Conjugacy classes for groups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20C20 | Modular representations and characters |
Software:
GAPReferences:
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