\(S\)-semipermutability of subgroups of \(p\)-nilpotent residual and \(p\)-supersolubility of a finite group. (English) Zbl 1478.20019
All groups considered in this report are finite. The aim of this paper is to obtain some conditions related with \(p\)-supersolubility of a group under the hypothesis that some subgroups of prime power order satisfy a certain subgroup embedding property. These results extend some other known results in the literature with stronger subgroup embedding properties.
A subgroup \(H\) of a group \(G\) is said to be S-permutable in \(G\) if it permutes with all Sylow subgroups of \(G\), and to be S-semipermutable in \(G\) if it permutes with all Sylow subgroups of \(G\) whose order is relatively prime to \(\lvert H\rvert\). We use the symbol \(\mathfrak{N}_p\) to denote the class of all \(p\)-nilpotent groups and \(G^{\mathfrak{N}_p}\) to denote the \(p\)-nilpotent residual of \(G\), that is, the smallest normal subgroup of \(G\) with \(p\)-nilpotent quotient. The first main result of this paper is Theorem 1.3: Let \(P\) be a Sylow \(p\)-subgroup of \(G\) and let \(d\) be a power of \(p\) such that \(p^2\le p\le \lvert P\rvert\). Assume that \(H\cap G^{\mathfrak{N}_p}\) is S-semipermutable in \(G\) for every noncyclic normal subgroup \(H\) of \(P\) with \(\lvert H\rvert=p\). Then either \(G\) is \(p\)-supersoluble, or else \(\lvert P\cap G^{\mathfrak{N}_p}\rvert >d\).
Given a prime \(p\), let us denote by \(G^*\) the smallest normal subgroup of \(G\) whose quotient is abelian of exponent dividing \(p-1\). Theorem 1.4 states that if \(P\) is a Sylow \(p\)-subgroup of \(G\) and \(d\) is a power of \(p\) such that \(p^2\le d<\lvert P\rvert\) and \(H\cap (G^*)^{\mathfrak{N}_p}\) is S-permutable in \(G\) for every noncyclic normal subgroup \(H\) of \(P\) with \(\lvert H\rvert =d\), then either \(G\) is \(p\)-supersoluble, or else \(\lvert P\cap (G^*)^{\mathfrak{N}_p}\rvert >d\).
Consider a fixed prime \(p\). A group \(G\) is said to be \(p\)-quasinilpotent if \(G\) induces inner automorphisms on all its chief factors of order divisible by \(p\). We denote by \(F_p^*(G)\) the \(p\)-quasinilpotent radical of a group \(G\), that is, the product of all normal subgroups of \(G\) that are \(p\)-quasinilpotent. The \(p\)-supersoluble hypercentre of \(G\) is the product of all normal subgroups \(T\) of \(G\) such that the \(p\)-\(G\)-chief factors of \(H\) have order \(p\). For a normal subgroup \(E\) of \(G\) and a prime \(p\) dividing \(\lvert G\rvert\), Theorem 1.5 affirms that if there exist a normal subgroup \(X\) of \(G\) such that \(F^*_p(E)\le X\le E\) and a noncyclic Sylow \(p\)-subgroup \(P\) of \(X\) such that \(H\cap (G^*)^{\mathfrak{N}_p}\) is S-semipermutable in \(G\) for every noncyclic subgroup \(H\) of \(P\) with \(\lvert H\rvert=d\), where \(d\) is a power of \(p\) with \(p^3\le d<\lvert P\rvert\), then \(E\) is contained in the \(p\)-supersoluble hypercentre of \(G\).
A subgroup \(H\) of a group \(G\) is said to be S-permutable in \(G\) if it permutes with all Sylow subgroups of \(G\), and to be S-semipermutable in \(G\) if it permutes with all Sylow subgroups of \(G\) whose order is relatively prime to \(\lvert H\rvert\). We use the symbol \(\mathfrak{N}_p\) to denote the class of all \(p\)-nilpotent groups and \(G^{\mathfrak{N}_p}\) to denote the \(p\)-nilpotent residual of \(G\), that is, the smallest normal subgroup of \(G\) with \(p\)-nilpotent quotient. The first main result of this paper is Theorem 1.3: Let \(P\) be a Sylow \(p\)-subgroup of \(G\) and let \(d\) be a power of \(p\) such that \(p^2\le p\le \lvert P\rvert\). Assume that \(H\cap G^{\mathfrak{N}_p}\) is S-semipermutable in \(G\) for every noncyclic normal subgroup \(H\) of \(P\) with \(\lvert H\rvert=p\). Then either \(G\) is \(p\)-supersoluble, or else \(\lvert P\cap G^{\mathfrak{N}_p}\rvert >d\).
Given a prime \(p\), let us denote by \(G^*\) the smallest normal subgroup of \(G\) whose quotient is abelian of exponent dividing \(p-1\). Theorem 1.4 states that if \(P\) is a Sylow \(p\)-subgroup of \(G\) and \(d\) is a power of \(p\) such that \(p^2\le d<\lvert P\rvert\) and \(H\cap (G^*)^{\mathfrak{N}_p}\) is S-permutable in \(G\) for every noncyclic normal subgroup \(H\) of \(P\) with \(\lvert H\rvert =d\), then either \(G\) is \(p\)-supersoluble, or else \(\lvert P\cap (G^*)^{\mathfrak{N}_p}\rvert >d\).
Consider a fixed prime \(p\). A group \(G\) is said to be \(p\)-quasinilpotent if \(G\) induces inner automorphisms on all its chief factors of order divisible by \(p\). We denote by \(F_p^*(G)\) the \(p\)-quasinilpotent radical of a group \(G\), that is, the product of all normal subgroups of \(G\) that are \(p\)-quasinilpotent. The \(p\)-supersoluble hypercentre of \(G\) is the product of all normal subgroups \(T\) of \(G\) such that the \(p\)-\(G\)-chief factors of \(H\) have order \(p\). For a normal subgroup \(E\) of \(G\) and a prime \(p\) dividing \(\lvert G\rvert\), Theorem 1.5 affirms that if there exist a normal subgroup \(X\) of \(G\) such that \(F^*_p(E)\le X\le E\) and a noncyclic Sylow \(p\)-subgroup \(P\) of \(X\) such that \(H\cap (G^*)^{\mathfrak{N}_p}\) is S-semipermutable in \(G\) for every noncyclic subgroup \(H\) of \(P\) with \(\lvert H\rvert=d\), where \(d\) is a power of \(p\) with \(p^3\le d<\lvert P\rvert\), then \(E\) is contained in the \(p\)-supersoluble hypercentre of \(G\).
Reviewer: Ramón Esteban-Romero (València)
MSC:
| 20D40 | Products of subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
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