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Register a new userFinite groups can be generated by a pi-subgroup and a pi-subgroup
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Answering a question of Dan Haran and generalizing some results of Aschbacher-Guralnick and Suzuki, we prove that given a set of primes pi, any finite group can be generated by a pi-subgroup and a pi-subgroup. This gives a free product description of a countably generated free profinite group.
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The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G}to Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gammaright)$? Here $hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Gamma)$ we denote $Cleft(Gammaright)=Cleft(Aut(Gamma),Gammaright)$. Let $Gamma$ be a finitely generated group, $bar{Gamma}=Gamma/[Gamma,Gamma]$, and $Gamma^{*}=bar{Gamma}/tor(bar{Gamma})congmathbb{Z}^{(d)}$. Denote $Aut^{*}(Gamma)=textrm{Im}(Aut(Gamma)to Aut(Gamma^{*}))leq GL_{d}(mathbb{Z})$. In this paper we show that when $Gamma$ is nilpotent, there is a canonical isomorphism $Cleft(Gammaright)simeq C(Aut^{*}(Gamma),Gamma^{*})$. In other words, $Cleft(Gammaright)$ is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group $Aut^{*}(Gamma)$. In particular, in the case where $Gamma=Psi_{n,c}$ is a finitely generated free nilpotent group of class $c$ on $n$ elements, we get that $C(Psi_{n,c})=C(mathbb{Z}^{(n)})={e}$ whenever $ngeq3$, and $C(Psi_{2,c})=C(mathbb{Z}^{(2)})=hat{F}_{omega}$ = the free profinite group on countable number of generators.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $mathrm{FP}_infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.
Let ${frak F}$ be a class of group and $G$ a finite group. Then a set $Sigma $ of subgroups of $G$ is called a emph{$G$-covering subgroup system} for the class ${frak F}$ if $Gin {frak F}$ whenever $Sigma subseteq {frak F}$. We prove that: {sl If a set of subgroups $Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Sigma$ is a $G$-covering subgroup system for the classes of all $sigma$-soluble and all $sigma$-nilpotent groups, and for the class of all $sigma$-soluble $Psigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.
A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndons axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumanns theorem. We study also the CSA property in such groups.
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