Affine semi-linear groups with three irreducible character degrees. (English) Zbl 1006.20004
Consider the following Hypothesis (*): Let a finite group \(G\) act faithfully on an elementary Abelian \(p\)-group \(V\), where \(p\) is a prime. Let \(A\) be maximal among Abelian normal subgroups of \(G\) and suppose that \(A\) acts irreducibly on \(V\).
Set \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). The following results are proved. Theorem. Let \(G\), \(V\), and \(A\) satisfy Hypothesis (*). Let \(|V|=q^n\), where \(q>1\) is a prime power, \(n>1\). Then (a) Suppose that \(GV\) is a Frobenius group and \(\text{cd}(GV)=\{1,n,|G|\}\) for appropriate \(G\), \(V\), and \(A\) with \(n=|G:A|\) if and only if every prime divisor of \(n\) divides \(q-1\). (b) Suppose that \(|\text{cd}(GV)|=3\). Then either \(GV\) is a Frobenius group and \(\text{cd}(GV)=\{1,|G:A|,|G|\}\), or \(|G:A|\) is a power of a prime \(r\), and \(\text{cd}(GV)=\{1,|G:A|,|G|/r\}\). In that case either \(|G:A|=r\) or \(r^2\) divides \(q-1\). (c) If \(\text{cd}(G)=\{1,a,b\}\) for integers \(1<a<b\), then either \(a\) divides \(b\) or else \(a\) and \(b\) are coprime.
Note that Lemma 2.5 in a paper by T. Noritzsch [J. Algebra 175, No. 3, 767-798 (1995; Zbl 0839.20014)] is incorrect as one can see from part (a) of the theorem.
Set \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). The following results are proved. Theorem. Let \(G\), \(V\), and \(A\) satisfy Hypothesis (*). Let \(|V|=q^n\), where \(q>1\) is a prime power, \(n>1\). Then (a) Suppose that \(GV\) is a Frobenius group and \(\text{cd}(GV)=\{1,n,|G|\}\) for appropriate \(G\), \(V\), and \(A\) with \(n=|G:A|\) if and only if every prime divisor of \(n\) divides \(q-1\). (b) Suppose that \(|\text{cd}(GV)|=3\). Then either \(GV\) is a Frobenius group and \(\text{cd}(GV)=\{1,|G:A|,|G|\}\), or \(|G:A|\) is a power of a prime \(r\), and \(\text{cd}(GV)=\{1,|G:A|,|G|/r\}\). In that case either \(|G:A|=r\) or \(r^2\) divides \(q-1\). (c) If \(\text{cd}(G)=\{1,a,b\}\) for integers \(1<a<b\), then either \(a\) divides \(b\) or else \(a\) and \(b\) are coprime.
Note that Lemma 2.5 in a paper by T. Noritzsch [J. Algebra 175, No. 3, 767-798 (1995; Zbl 0839.20014)] is incorrect as one can see from part (a) of the theorem.
Reviewer: Yakolev Berkovich (Afula)
MSC:
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Citations:
Zbl 0839.20014References:
| [1] | Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.07201 |
| [2] | Isaacs, I. M., Algebra: A Graduate Course (1994), Brooks/Cole: Brooks/Cole Pacific Grove · Zbl 0805.00001 |
| [3] | Isaacs, I. M., Character Theory of Finite Groups (1994), Dover: Dover New York · Zbl 0823.20005 |
| [4] | Manz, O.; Wolf, T. R., Representations of Solvable Groups (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0928.20008 |
| [5] | Noritzsch, T., Groups having three complex irreducible character degrees, J. Algebra, 175, 767-798 (1995) · Zbl 0839.20014 |
| [6] | Roitman, M., On Zsigmondy primes, Proc. Amer. Math. Soc., 125, 1913-1919 (1997) · Zbl 0914.11002 |
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