Narrow Results

A cover for a finite group is a set of proper subgroups whose union is the whole group. For a finite group G, we denote by σ (G) the least positive integer n such that G has a cover of size n. This paper deals with the covers of a finite soluble group G having σ (G) elements of maximal subgroups.

We prove that every free metabelian non-cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary, we prove that for every prime number p, an arbitrary free metabelian non-cyclic group has a finitely generated p′-isolated subgroup which is not p-separable.

In this paper we completely study the nilpotency and solubility of the group Aut_c(G) of central automorphisms of a finite group G.

The norm N(G) of a group G is the intersection of the normalizes of all the subgroups of G. A group is called capable if it is a central factor group. In this paper, we give a necessary and sufficient condition for a capable group to satisfy N(G)=ζ(G), and then some sufficient conditions for a capable group with N(G)=ζ(G) are obtained. Furthermore, we discuss the norm of a nilpotent group with cyclic derived subgroup.

Let be a saturated formation containing , and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) . (ii) There exists a normal subgroup H of G such that and for each Sylow subgroup P of F*(H), every member in some satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed on the structure of G.

Finite nilpotent groups having exactly four conjugacy classes of non-normal subgroups are classified.

Let $A$ be a completely decomposable homogeneous torsion-free abelian group of rank $n$ ($n\ge 2$). Let ${\mathrm{\Theta }}_m(n)=A⋊⟨\alpha ⟩$ be the split extension of $A$ by an automorphism $\alpha$ which is a cyclic permutation of the direct components twisted by a rational integer $m$. Then ${\mathrm{\Theta }}_m(n)$ is an infinite soluble group. In this paper, the residual finiteness of ${\mathrm{\Theta }}_m(n)$ is investigated.

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup $\mu$ of $P$ and ${\beta }^{*}$, the fundamental relation on a given polygroup $P$, we construct an Engel fuzzy subgroup ${\mu }_{{\beta }^{*}}$. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group $P$/$\sim$, the group of equivalence classes derived from a fuzzy subpolygroup of $P$. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

Let $G$ be a group given by a free presentation $G\cong F/R$. The 2-nilpotent multiplier of $G$ is the abelian group ${𝕄}^{(2)}(G)=(R\cap {\gamma }_3(F))/{\gamma }_3(R,F)$ which is invariant of $G$ [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc.58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z.226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of ${𝕄}^{(2)}(G)$, when $G$ is a finite (generalized) extra special $p$-group. Moreover, the descriptions of the triple tensor product ${\otimes }^{3}G$, and the triple exterior product ${\wedge }^{3}G$ are given.

Let H, C and D be subgroups of a finite group G. We call the pair (C, D) a θ*-pair for H if it satisfies the following conditions: (i) D = (C∩H)_G, 〈C^g, H〉 = G for every g ∈ G, (ii) KH < G for every K/D ◃ G/D with K/D < C/D. In this paper, we obtain several results on the maximal θ*-pair which implies G to be solvable, supersolvable and nilpotent.

In this survey we show the influence of the abnormal maximal subgroups of finite groups in their structure.

In this paper we discuss some constructions and recent results of the theory of soluble and partially soluble finite groups.