An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups. (English) Zbl 0592.20006
It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.
We present the following theorems: Theorem A. If C and D are non-trivial conjugacy classes of a finite group G such that either \(CD=mC+nD\) or \(CD=mC^{-1}+nD\), where m,n are non-negative integers, then G is not a non-abelian simple group. - Theorem B. If \(\chi\) and \(\psi\) are non- trivial irreducible characters of a finite group G such that either \(\chi \psi =m\chi +n\psi\) or \(\chi \psi =m{\bar \chi}+n\psi\), where m,n are non-negative integers, then G is not a non-abelian simple group.
We present the following theorems: Theorem A. If C and D are non-trivial conjugacy classes of a finite group G such that either \(CD=mC+nD\) or \(CD=mC^{-1}+nD\), where m,n are non-negative integers, then G is not a non-abelian simple group. - Theorem B. If \(\chi\) and \(\psi\) are non- trivial irreducible characters of a finite group G such that either \(\chi \psi =m\chi +n\psi\) or \(\chi \psi =m{\bar \chi}+n\psi\), where m,n are non-negative integers, then G is not a non-abelian simple group.
MSC:
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
References:
| [1] | Isaacs, Character Theory of Finite Groups (1976) · Zbl 0337.20005 |
| [2] | DOI: 10.1016/0021-8693(86)90183-3 · Zbl 0598.20002 · doi:10.1016/0021-8693(86)90183-3 |
| [3] | DOI: 10.1007/BFb0072289 · doi:10.1007/BFb0072289 |
| [4] | Arad, Products of Conjugacy Classes in Groups (1985) · Zbl 0561.20004 · doi:10.1007/BFb0072284 |
| [5] | DOI: 10.1007/BFb0072286 · doi:10.1007/BFb0072286 |
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