## 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
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### References:

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