Abelian quasinormal subgroups of groups.

*(English)*Zbl 1154.20306
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 15, No. 2, 69-79 (2004).

From the introduction: Among the most important concepts in group theory, arguably the most important, are those of composition and chief series, arising from normal subgroups. If ‘normal’ is replaced by ‘quasinormal’, then, by analogy, little seems to be known. A subgroup \(A\) of a group \(G\) is said to be ‘quasinormal’ (or ‘permutable’) in \(G\) if \(AX=XA\) for all subgroups \(X\) of \(G\). Obviously this is equivalent to the product \(AX\) being a subgroup. Of course normal subgroups are necessarily quasinormal, while the converse is not always true. But what is the structure of minimal quasinormal subgroups and what properties do maximal (i.e. unrefinable) chains of quasinormal subgroups possess? The answers to these and similar questions remain pitifully inadequate. In the present work, however, we show that when \(A\) is an Abelian quasinormal subgroup of \(G\), then certain canonical subgroups of \(A\) are also quasinormal in \(G\). We denote the core of a subgroup \(A\) of a group \(G\) by \(A_G\). When \(A\) is quasinormal and \(G\) is finite, the quotient \(A/A_G\) is always nilpotent [N. Itô, J. Szép, Acta Sci. Math. 23, 168-170 (1962; Zbl 0112.02106)], though there is no restriction on the class [F. Gross, Rocky Mt. J. Math. 1, 541-550 (1971; Zbl 0238.20040), S. E. Stonehewer, J. Aust. Math. Soc. 16, 90-97 (1973; Zbl 0279.20017)]. Much of the published work on quasinormal subgroups has been about \(A/A_G\), i.e. the core-free case. Here, however, our arguments apply equally to the core-free and the noncore-free situations.

Our main result is the following Theorem 1. Let \(G\) be any group and let \(A\) be an Abelian quasinormal subgroup of \(G\). If \(n\) is any positive integer, either odd or divisible by 4, then we prove that the subgroup \(A^n\) is also quasinormal in \(G\).

We see in Section 3 that the restriction on \(n\) here is necessary, by constructing an example in which \(A^2\) is not quasinormal. Theorem 1 is proved first for the case when \(G\) is a finite group. This in turn reduces easily to the case where \(G\) is a \(p\)-group, for some prime \(p\). Then by straightforward induction arguments, we need to consider only the cases \(n=p\), when \(p\) is odd, and \(n=4\) and 8, when \(p=2\).

Section 4 deals with infinite groups \(G\) and deduces the full statement of Theorem 1 from the finite case. Also we include here two further examples which answer obvious questions.

Our main result is the following Theorem 1. Let \(G\) be any group and let \(A\) be an Abelian quasinormal subgroup of \(G\). If \(n\) is any positive integer, either odd or divisible by 4, then we prove that the subgroup \(A^n\) is also quasinormal in \(G\).

We see in Section 3 that the restriction on \(n\) here is necessary, by constructing an example in which \(A^2\) is not quasinormal. Theorem 1 is proved first for the case when \(G\) is a finite group. This in turn reduces easily to the case where \(G\) is a \(p\)-group, for some prime \(p\). Then by straightforward induction arguments, we need to consider only the cases \(n=p\), when \(p\) is odd, and \(n=4\) and 8, when \(p=2\).

Section 4 deals with infinite groups \(G\) and deduces the full statement of Theorem 1 from the finite case. Also we include here two further examples which answer obvious questions.

##### MSC:

20E07 | Subgroup theorems; subgroup growth |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20E22 | Extensions, wreath products, and other compositions of groups |

20D40 | Products of subgroups of abstract finite groups |

20D15 | Finite nilpotent groups, \(p\)-groups |

20F14 | Derived series, central series, and generalizations for groups |