Non-solvable groups whose non-linear character degrees have the same number of different prime divisors
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908954751139840 |
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| author | Guo, Junying Liu, Yanjun Wu, Ziyi Xiao, Di |
| author_facet | Guo, Junying Liu, Yanjun Wu, Ziyi Xiao, Di |
| contents | By a result of Noritzsch, a finite solvable group whose non-linear character degrees have the same set of prime divisors is meta-abelian. In this note we investigate finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors, and show that up to an abelian direct factor, such groups are exactly $L_2(4), L_2(8), A_7, S_7$, the central product of a cyclic $3$-group with $3.A_7$, or the semi-direct product of $A_7$ by a cyclic $2$-group $\langle a\rangle$ such that $a$ non-trivially acts on $A_7$ by conjugation. As consequence, we show that only the primes $2,3,5,7$ may occur as prime divisors of their irreducible character degrees, and that Huppert's $ρ$-$σ$ conjecture holds for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-solvable groups whose non-linear character degrees have the same number of different prime divisors Guo, Junying Liu, Yanjun Wu, Ziyi Xiao, Di Representation Theory Group Theory 20C15, 20D05 By a result of Noritzsch, a finite solvable group whose non-linear character degrees have the same set of prime divisors is meta-abelian. In this note we investigate finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors, and show that up to an abelian direct factor, such groups are exactly $L_2(4), L_2(8), A_7, S_7$, the central product of a cyclic $3$-group with $3.A_7$, or the semi-direct product of $A_7$ by a cyclic $2$-group $\langle a\rangle$ such that $a$ non-trivially acts on $A_7$ by conjugation. As consequence, we show that only the primes $2,3,5,7$ may occur as prime divisors of their irreducible character degrees, and that Huppert's $ρ$-$σ$ conjecture holds for them. |
| title | Non-solvable groups whose non-linear character degrees have the same number of different prime divisors |
| topic | Representation Theory Group Theory 20C15, 20D05 |
| url | https://arxiv.org/abs/2604.10100 |