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Character degree graphs and normal subgroups. (English) Zbl 1034.20009
If $$Z$$ is a subset of $$\mathbb{N}$$, then the ‘common-divisor graph’ $${\mathcal G}(Z)$$ of $$Z$$ has vertex set $$Z$$, and distinct vertices $$m$$ and $$n$$ are joined if $$\text{GCD}(m,n)>1$$. For $$N\trianglelefteq G$$, set $$\text{Irr}(G| N)=\{\chi\in\text{Irr}(G)\mid N\nleq\ker(\chi)\}$$, $$\text{cd}(G| N)=\{\chi(1)\mid\chi\in\text{Irr}(G| N)\}$$. Next, denote by $$c(\mathcal G)$$ the number of connected components of a graph $$\mathcal G$$. We retain the notation so introduced.
The author studies the relative degree graph $${\mathcal G}(G| N)={\mathcal G}(\text{cd}(G| N))$$ in great detail. Below the main results are formulated. In what follows $$N\leq G'$$ is a $$G$$-invariant subgroup of $$G$$ and $${\mathcal G}={\mathcal G}(G/N)$$, $$c=c(\mathcal G)$$.
(a) If $$N'<N$$, then $$c\leq 2$$. If $$c=2$$ and $${\mathcal X}_1$$ and $${\mathcal X}_2$$ are connected components of our graph, then all members of $${\mathcal X}_1$$ are coprime to $$| N:N'|$$ and $$\pi(m)=\pi(| N:N'|)$$ for all $$m\in{\mathcal X}_2$$.
(b) Let $$N$$ be solvable and $$c=2$$. Then the nilpotent length of $$G$$ is at most $$3$$. Also, $$N''$$ is either nilpotent or $$G''$$ contains a $$G$$-invariant Abelian subgroup of index $$2$$.
(c) Let $$N$$ be an Abelian $$p$$-group and $$P\in\text{Syl}_p(G)$$. If $$c=2$$, then (i) $$P$$ is normal in $$G$$ and $$G/P$$ and $$P/N$$ are Abelian; (ii) $$N$$ is not central in $$P$$, but every minimal normal subgroup of $$G$$ contained in $$G'$$ is central in $$G$$.
(d) If $$N\leq G'$$ is an Abelian minimal normal subgroup of $$G$$, then $$c=1$$.
(e) Let $$N$$ be solvable and $$c=2$$. Then one component of $$\mathcal G$$ is a complete graph and the other component has diameter $$2$$.
(f) If $$N$$ is a $$p$$-group, then no two vertices of $$\mathcal G$$ can have finite distance exceeding $$3$$. – The proof of (f) is very involved.
This important paper contains a number of interesting related results. It develops some ideas of a paper of I. M. Isaacs and G. Knutson [J. Algebra 199, No. 1, 302-326 (1998; Zbl 0889.20005)].

##### MSC:
 20C15 Ordinary representations and characters 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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