Groups with two extreme character degrees and their normal subgroups.(English)Zbl 0968.20005

Let $$G$$ be a finite group and $$\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}$$. The following results are proven.
Theorem A. For a non-Abelian $$p$$-group $$G$$ the following conditions are equivalent: (i) $$\text{cd}(G)=\{1,|G:Z(G)|^{1/2}\}$$. (ii) The set of conjugacy class sizes of $$G$$ is $$\{1,|G'|\}$$. (iii) $$G'=[x,G]$$ for all $$x\in G-Z(G)$$. (iv) $$Z(G/N)=Z(G)/N$$ for any normal subgroup $$N$$ of $$G$$ such that $$G'\nleq N$$.
Theorem B. Let $$G$$ be a $$p$$-group such that $$|G:Z(G)|=p^{2n}$$ is a square. Then the following statements are equivalent: (i) $$\text{cd}(G)=\{1,p^n\}$$. (ii) The normal subgroups of $$G$$ either contain $$G'$$ or are contained in $$Z(G)$$.
Theorem C. Let $$G$$ be a $$p$$-group such that $$|G:Z(G)|=p^{2n+1}$$ is not a square. Then the following statements are equivalent: (i) $$\text{cd}(G)=\{1,p^n\}$$. (ii) For any $$N\trianglelefteq G$$, either $$G'\leq N$$ or $$|NZ(G):Z(G)|\leq p$$.
Some related results are also presented.

MSC:

 20C15 Ordinary representations and characters 20D15 Finite nilpotent groups, $$p$$-groups 20E45 Conjugacy classes for groups 20D30 Series and lattices of subgroups 20D60 Arithmetic and combinatorial problems involving abstract finite groups

GAP
Full Text:

References:

 [1] Wolfgang Bannuscher, Über Gruppen mit wenigen irreduziblen Charakteren. I, II, Math. Nachr. 153 (1991), 79 – 84, 131 – 135 (German). · Zbl 0806.20007 [2] Wolfgang Bannuscher, Über Gruppen mit genau zwei irreduziblen Charaktergraden. I, II, Math. Nachr. 154 (1991), 253 – 258, 259 – 263 (German). · Zbl 0752.20004 [3] Bert Beisiegel, Semi-extraspezielle \?-Gruppen, Math. Z. 156 (1977), no. 3, 247 – 254 (German). · Zbl 0346.20016 [4] Rex Dark and Carlo M. Scoppola, On Camina groups of prime power order, J. Algebra 181 (1996), no. 3, 787 – 802. · Zbl 0860.20017 [5] L. Di Martino and M. C. Tamburini, Some remarks on the degrees of faithful irreducible representations of a finite \?-group, Geom. Dedicata 41 (1992), no. 2, 155 – 164. · Zbl 0789.20004 [6] Marshall Hall Jr. and James K. Senior, The groups of order 2$$^{n}$$(\?\le 6), The Macmillan Co., New York; Collier-Macmillan, Ltd., London, 1964. · Zbl 0192.11701 [7] P. HALL, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141. · JFM 66.0081.01 [8] Hermann Heineken, Nilpotent groups of class two that can appear as central quotient groups, Rend. Sem. Mat. Univ. Padova 84 (1990), 241 – 248 (1991). · Zbl 0722.20011 [9] Robert B. Howlett and I. Martin Isaacs, On groups of central type, Math. Z. 179 (1982), no. 4, 555 – 569. · Zbl 0511.20002 [10] B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). · Zbl 0217.07201 [11] Bertram Huppert, Character theory of finite groups, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. · Zbl 0932.20007 [12] I.M. ISAACS, “Character Theory of Finite Groups”, Dover, New York, 1994. · Zbl 0849.20004 [13] I. M. Isaacs and D. S. Passman, A characterization of groups in terms of the degrees of their characters, Pacific J. Math. 15 (1965), 877 – 903. · Zbl 0132.01902 [14] I. M. Isaacs and D. S. Passman, A characterization of groups in terms of the degrees of their characters. II, Pacific J. Math. 24 (1968), 467 – 510. · Zbl 0155.05502 [15] Brian G. Wybourne, Rank dependency of group properties, Proceedings of the Tenth International Colloquium on Group-Theoretical Methods in Physics (Canterbury, 1981), 1982, pp. 350 – 360. · Zbl 0511.22018 [16] E. I. Khukhro, \?-automorphisms of finite \?-groups, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. · Zbl 0897.20018 [17] Avinoam Mann, Minimal characters of \?-groups, J. Group Theory 2 (1999), no. 3, 225 – 250. · Zbl 0940.20014 [18] Thomas Noritzsch, Groups having three complex irreducible character degrees, J. Algebra 175 (1995), no. 3, 767 – 798. · Zbl 0839.20014 [19] M. SCHÖNERT ET AL., “GAP - Groups, Algorithms, and Programming”, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, fifth edition, 1995. [20] Libero Verardi, Semi-extraspecial groups of exponent \?, Ann. Mat. Pura Appl. (4) 148 (1987), 131 – 171 (Italian, with English summary). · Zbl 0648.20032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.