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We give a method to describe all congruence images of a finitely generated Zariski dense group $H leq mathrm{SL}(n, mathbb{Z})$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we compute all congruence images only modulo primes. We propose a separate method that works for all $n$ as long as $H$ contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.
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Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.
For $n > 2$, let $Gamma$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $Hleq Gamma$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $Gamma$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.
Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is s-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized.
In this paper, we study a group in which every 2-maximal subgroup is a Hall subgroup.
A finite group $P$ is said to be emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-group. In this paper, we determine the structure of a finite group $G$ in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of $G$ satisfies the Frobenius condition, then $G/F(G)$ is cyclic and every maximal non-normal nilpotent subgroup $U$ of $G$ with $F(G)U=G$ is a Carter subgroup of $G$.
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