Character degree graphs and normal subgroups.

*(English)*Zbl 1034.20009If \(Z\) 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)].

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)].

Reviewer: Yakov Berkovich (Afula)

##### 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.) |

##### Keywords:

common-divisor graphs; irreducible characters; connected components; character degrees; finite solvable groups; nilpotent lengths; normal subgroups
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\textit{I. M. Isaacs}, Trans. Am. Math. Soc. 356, No. 3, 1155--1183 (2004; Zbl 1034.20009)

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