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Let G be a group and DS(G) = { H| H is any subgroup of G}. G is said to be a DC-group if DS(G) is a chain. In this paper, we prove that a finite DC-group is a semidirect product of a Sylow p-subgroup and an abelian p-subgroup. For the case of G being a finite p-group, we obtain some properties of a DC-group. In particular, a DC 2-group is characterized. Moreover, we prove that DC-groups are metabelian for p<5 and give an example that a non-abelian DC-group is not be necessarily metabelian for p>3.
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In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
A theorem of Dolfi, Herzog, Kaplan, and Lev cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in cite{DHKL}, we establish the following generalisation: if $mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $overline{mathfrak{X}}$ is the extension-closure of $mathfrak{X}$, then there exists an (optimal) constant $gamma$ depending only on $mathfrak{X}$ such that, for all non-trivial finite groups $G$ with trivial $mathfrak{X}$-radical, begin{equation} leftlvert G^{overline{mathfrak{X}}}rightrvert ,>, vert Gvert^gamma, end{equation} where $G^{overline{mathfrak{X}}}$ is the ${overline{mathfrak{X}}}$-residual of $G$. When $mathfrak{X} = mathfrak{N}$, the class of finite nilpotent groups, it follows that $overline{mathfrak{X}} = mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.,G. Thompsons classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations $mathfrak{X}$ of full characteristic such that $mathfrak{S} subset overline{mathfrak{X}} subset mathfrak{E}$, thus providing applications of our main result beyond the reach of cite[Thm.~C]{DHKL}.
Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.
Finite symmetries abound in particle physics, from the weak doublets and generation triplets to the baryon octet and many others. These are usually studied by starting from a Lie group, and breaking the symmetry by choosing a particular copy of the Weyl group. I investigate the possibility of instead taking the finite symmetries as fundamental, and building the Lie groups from them by means of a group algebra construction.
Let $G$ be a finite group admitting a coprime automorphism $alpha$ of order $e$. Denote by $I_G(alpha)$ the set of commutators $g^{-1}g^alpha$, where $gin G$, and by $[G,alpha]$ the subgroup generated by $I_G(alpha)$. We study the impact of $I_G(alpha)$ on the structure of $[G,alpha]$. Suppose that each subgroup generated by a subset of $I_G(alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(alpha)$ has odd order, then $[G,alpha]$ has odd order too. Further, if every pair of elements from $I_G(alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,alpha]$ is soluble, or respectively nilpotent.
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