Notes on π-Quasi-Normal Subgroups in Finite Groups

Abstract

Let G be a finite group and let it be a set of primes. A subgroup H of G is called 7r-quasi-normal in G if H permutes with every Sylow p-subgroup of G for every p in i . In this paper, we investigate how 7r-quasinormality conditions on some subgroups of G affect the structure of G. All groups considered are finite. The purpose of this paper is to investigate the influence of it-quasi-normality conditions on some subgroups of a finite group. it denotes a set of primes. Let G be a group, and let H be a subgroup of G. H is called 7:-quasi-normal in G if H permutes with every Sylowp-subgroup of G for every p in it; H is called' S-quasi-normal in G if H permutes with every Sylow subgroup of G; H is called quasi-normal in G if H permutes with every subgroup of G. Lemma 1. Let A, B, and C be subgroups of the group G. If A and B permute with C, then the subgroup (A, B) permutes with C [1, Hilfssatz 1, p. 207]. Lemma 2. Assume A < M < G and N a G. If A is it-quasi-normal in G, then A is 7r-quasi-normal in M and AN/N is it-quasi-normal in G/N [1, Hilfssatz 3, p. 207]. Lemma 3. Let A and B be subgroups in the group G. If A is it-quasi-normal in G and AB = BA, then A n B is it-quasi-normal in B [1, Hilfssatz 4, p. 207]. Lemma 4. Let A be a subgroup in the group G. If A is S-quasi-normal in G, then AaaiG [1, Satz l, p. 209]. Lemma 5. Assume A < G and P e Sylp (G) for every prime in p in it. If A is it-quasi-normal in G, then A n P e Sylp(A). Proof. Since AP is a subgroup of G, by the Sylow theorems, it follows that AnP e Sylp(A). Lemma 6. Let A be a maximal it-quasi-normal subgroup in G. Then one of the following statements is true: (a) A is a maximal normal subgroup in G. Received by the editors June 6, 199^0 and, in revised from, July 2, 1991. 1991 Mathematics Subject Classification. Primary 20D99, 20D40, 20F1 7; Secondary 20F1 6.