User poove - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:54:20Z http://mathoverflow.net/feeds/user/13835 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59209/hopfian-property Hopfian property Poove 2011-03-22T17:45:12Z 2012-07-22T22:21:40Z <p>Let $G$ be a group which is Hopfian and given a short exact sequence $1\to F \to H \to G \to 1$ with $F$ a finite normal subgroup of $H$. Is $H$ Hopfian? </p> http://mathoverflow.net/questions/83138/git-quotient-of-an-algebraic-action GIT quotient of an algebraic action Poove 2011-12-10T19:14:29Z 2011-12-10T20:40:09Z <p>Let $G$ be a connected affine algebraic group over an algebraically closed field $K$ which acts algebraically on an affine $K$-variety $V:=\bigcup_{i=1}^{n} V_i$, where $V_i$'s are irreducible $G$-stable pairwise (disjoint) isomorphic as varieties. If one of the restriction $\pi_i:V_i\rightarrow V_i//G$ of the quotient morphism $\pi:V\rightarrow V//G$ is constant, does that imply any $\pi_j:V_j\rightarrow V_j//G, \forall j\ne i,$ is also constant? ($V//G$ is the GIT quotient of $V$).</p> <p>It may be a trivial question, but not clear to me and I couldn't find out a counter example ( I believe its not true).</p> http://mathoverflow.net/questions/74634/suppose-gamma-m-is-a-principal-congruence-subgroup-of-level-m-contained-in-a-f Suppose $\Gamma_m$ is a principal congruence subgroup of level m contained in a finite index subgroup $\Gamma$ of $SL(n,\mathbb Z)$. Is $\Gamma_m$ characteristic in $\Gamma$? Poove 2011-09-06T07:00:04Z 2011-09-16T19:17:33Z <p>We know that principal congruence subgroups are characteristic in $SL(n,\mathbb Z)$. Suppose $\Gamma$ is a finite index subgroup of $SL(n,\mathbb Z)$ and $\Gamma_m$ is a principal congruence subgroup of level m contained in $\Gamma$. Will it be characteristic in $\Gamma$?</p> http://mathoverflow.net/questions/63319/principal-congruence-subgroups-of-sln-z Principal congruence subgroups of SL(n,Z) Poove 2011-04-28T18:27:32Z 2011-04-29T10:12:44Z <p><a href="http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz" rel="nofollow">http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz</a> gives a discription of automorphisms of $SL(n,\mathbb Z)$. Is it true for n even too?. Hau-Reiner's paper gives generators for the group of automorphisms of $SL(n,\mathbb Z)$ which are induced by the automorphisms of $GL(n,\mathbb Z)$. </p> http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz Principal congruence subgroups of $SL(n, \mathbb{Z})$ Poove 2011-04-20T18:33:32Z 2011-04-20T20:16:07Z <p>I want to know whether the principal congruence subgroups of $SL(n, \mathbb{Z})$ are characteristic? please suggest me a reference.</p> http://mathoverflow.net/questions/83138/git-quotient-of-an-algebraic-action Comment by Poove Poove 2011-12-11T08:37:10Z 2011-12-11T08:37:10Z @ Ben Webster: You are right, but here I have fixed a $G$ action on $V$ and so on each $V_i\subset V$, the restricted action of $G$. And $\pi_i:=\pi|V_i$. http://mathoverflow.net/questions/83138/git-quotient-of-an-algebraic-action Comment by Poove Poove 2011-12-10T19:15:57Z 2011-12-10T19:15:57Z I had asked this one day back, but deleted for the question was unclear. http://mathoverflow.net/questions/74634/suppose-gamma-m-is-a-principal-congruence-subgroup-of-level-m-contained-in-a-f/74650#74650 Comment by Poove Poove 2011-09-08T21:09:47Z 2011-09-08T21:09:47Z @Igor Rivin: Any automorphism of $\Gamma$ extends(super rigidity) to an automorphism of $SL(n,\mathbb R)$, for n&gt;2. But it not necessarily restricts as an automorphism to $SL(n,\mathbb Z)$, instead it restricts to the normalizer of $\Gamma$ in $SL(n,\mathbb R)$, Which is again a proper finite index subgroup of $SL(n,\mathbb Z)$. So <a href="http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz" rel="nofollow" title="principal congruence subgroups of sln mathbbz">mathoverflow.net/questions/62446/&hellip;</a> doesn't say $\Gamma_m$ is characteristic in $\Gamma$. http://mathoverflow.net/questions/74634/suppose-gamma-m-is-a-principal-congruence-subgroup-of-level-m-contained-in-a-f/74675#74675 Comment by Poove Poove 2011-09-08T20:55:30Z 2011-09-08T20:55:30Z @Agol: we can simply take $A \in SL(2,\mathbb Q)$. Thanks for the answer. http://mathoverflow.net/questions/38806/proper-subgroup-of-gln-z-isomorphic-to-gln-z/38810#38810 Comment by Poove Poove 2011-05-05T15:43:55Z 2011-05-05T15:43:55Z $PGL(n,\mathbb Z)= PSL(n,\mathbb Z)$ for n odd only. http://mathoverflow.net/questions/38806/proper-subgroup-of-gln-z-isomorphic-to-gln-z/38810#38810 Comment by Poove Poove 2011-05-03T16:22:20Z 2011-05-03T16:22:20Z @Igor Belegradek: But for n even, the centre of both $SL(n,\mathbb Z)$ and $GL(n,\mathbb Z)$ is $\{I,-I\}$, thus if $PGL(n,\mathbb Z)= PSL(n,\mathbb Z)$, then by one of the isomorphism theorems it will imply that $GL(n,\mathbb Z)= SL(n,\mathbb Z)$. I guess i am not wrong. http://mathoverflow.net/questions/38806/proper-subgroup-of-gln-z-isomorphic-to-gln-z/38810#38810 Comment by Poove Poove 2011-05-02T11:34:46Z 2011-05-02T11:34:46Z @Igor Belegradek:(If I'm not wrong) You have mentioned in your answer above that $PGL(n,\mathbb Z)= PSL(n,\mathbb Z)$ for all $n&gt;2$. Is it true for all such n?. http://mathoverflow.net/questions/63319/principal-congruence-subgroups-of-sln-z/63390#63390 Comment by Poove Poove 2011-05-02T08:28:55Z 2011-05-02T08:28:55Z @Ralph: I couldn't see any mistake in your consideration. :). http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz/62457#62457 Comment by Poove Poove 2011-04-29T15:19:56Z 2011-04-29T15:19:56Z @Ralph: Thanks. http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz/62457#62457 Comment by Poove Poove 2011-04-29T09:31:57Z 2011-04-29T09:31:57Z See this for answer: <a href="http://mathoverflow.net/questions/63319/principal-congruence-subgroups-of-sln-z" rel="nofollow" title="principal congruence subgroups of sln z">mathoverflow.net/questions/63319/&hellip;</a> http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz/62457#62457 Comment by Poove Poove 2011-04-27T09:30:44Z 2011-04-27T09:30:44Z @Ralph: $\Gamma_n$ is characteristic. but what you said about the automorphism group of $SL(n,\mathbb Z)$ is true for n odd, because for n odd, all automorphisms of $SL(n,\mathbb Z)$ are induced by automorphisms of $GL(n,\mathbb Z)$(Hua-Reiner,theorem.3). Is it true if n even too? http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz/62457#62457 Comment by Poove Poove 2011-04-21T05:56:26Z 2011-04-21T05:56:26Z you are right Ralph http://mathoverflow.net/questions/62446/principal-congruence-subgroups-of-sln-mathbbz/62452#62452 Comment by Poove Poove 2011-04-21T05:55:18Z 2011-04-21T05:55:18Z Thanks a lot Richard. http://mathoverflow.net/questions/59209/hopfian-property Comment by Poove Poove 2011-03-23T10:07:24Z 2011-03-23T10:07:24Z @ ndkrempel: G has no normal subgroups isomorphic to a free abelian group of finite rank. (dont want to assume G is torsion free. http://mathoverflow.net/questions/59209/hopfian-property Comment by Poove Poove 2011-03-22T21:03:28Z 2011-03-22T21:03:28Z @ ndkrempel: why the math symbols are not readable here? did you mean the group G or H on which i would need stronger assumptions? sorry for disturbing