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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.
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 subgroup
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
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.
Denote by $ u_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ u_p(H)leq u_p(G)$ for $Hleq G$, however $ u_p(H)$ does not divide $ u_p(G)$ in general. In this paper we reduce the question whether $ u_p(H)$ divides $ u
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 inequ