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Register a new userTopological finite generation of compact open subgroups of universal groups
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In this paper we characterize the finite permutation groups $F<S_d$ on $d$ letters such that every compact open subgroup of the associated universal group $U(F)<{rm Aut} T_d$ is topologically finitely generated. Actually we show that in this case the groups are positively finitely generated.
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It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
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.
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_p(G)$ for every $Hleq G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We say that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.
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