User solovei - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:21:56Z http://mathoverflow.net/feeds/user/25762 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128865/can-one-characterize-amenable-groups-with-c-gx-cyclic-for-all-x-neq-1 Can one characterize amenable groups with $C_G(x)$ cyclic for all $x\neq 1$? solovei 2013-04-26T22:00:16Z 2013-05-07T14:20:05Z <p>Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic?</p> <p>For example, must they be solvable?</p> http://mathoverflow.net/questions/128918/why-didnt-finite-group-theorists-consider-groups-where-all-centralizers-of-non-i Why didn't finite group theorists consider groups where all centralizers of non-identity elements are solvable? solovei 2013-04-27T13:07:14Z 2013-04-27T20:12:07Z <p>From Wikipedia article on the Feit-Thompson Theorem proving a conjecture of Burnside that groups of odd order are solvable: "The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)</p> <p>Feit, Hall, and Thompson (1960) extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory."</p> http://mathoverflow.net/questions/128841/classification-of-groups-in-which-the-centralizer-of-every-non-identity-element-i Classification of groups in which the centralizer of every non-identity element is cyclic solovei 2013-04-26T17:18:22Z 2013-04-26T22:02:14Z <p>In which classes of groups is it feasible to classify those groups in which the centralizer of every non-identity element is cyclic?</p> http://mathoverflow.net/questions/123460/cubic-fields-up-to-isomorphism Cubic Fields Up to Isomorphism solovei 2013-03-03T06:35:32Z 2013-03-04T06:12:47Z <p>Why are there only finitely many cubic fields of a given discriminant? Is this true for higher dimension too? </p> <p>What other invariants are needed to classify cubic fields? number of real and complex embeddings, Galois closure?...</p> http://mathoverflow.net/questions/107902/finite-subgroups-of-sl-2r Finite Subgroups of $SL_2(R)$ solovei 2012-09-23T13:55:07Z 2012-09-25T15:06:15Z <p>Can you show any finite subgroup of $SL_2(R)$ is cyclic without using an invariant form?</p> http://mathoverflow.net/questions/107182/center-of-p-groups Center of p-groups solovei 2012-09-14T13:53:50Z 2012-09-15T03:56:51Z <p>Can one show that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p$?</p> http://mathoverflow.net/questions/107082/decision-problem-for-finitely-generated-subgroups Decision Problem for finitely generated subgroups solovei 2012-09-13T12:29:58Z 2012-09-13T20:48:26Z <p>Suppose $G$ is a finitely generated subgroup of $GL_n(Z)$, $n\ge 3$. I suspect that there is no decision procedure for deciding whether or not such $G$ is finitely presented. How can this proved? </p> http://mathoverflow.net/questions/106352/divisors-of-an-integer Divisors of an integer solovei 2012-09-04T14:50:35Z 2012-09-04T14:50:35Z <p>Say $M$ is the number of divisors of an integer. Is there a simple formula for the minimal integer $n$ so that the number of divisors of $n$ is $M$?</p> http://mathoverflow.net/questions/106049/euclidean-inside-hyperbolic Euclidean inside Hyperbolic solovei 2012-08-31T16:09:44Z 2012-09-01T08:00:17Z <p>One can make a model of the hyperbolic plane inside the Euclidean plane, either using the conformal model or projective model. </p> <p>How does one make a model of the Euclidean plane inside the hyperbolic plane? </p> http://mathoverflow.net/questions/104911/composition-series Composition Series solovei 2012-08-17T13:00:50Z 2012-08-18T08:15:15Z <p>Which finite groups have uniqueness for the ordered sequence of composition factors (up to isomorphism)?</p> http://mathoverflow.net/questions/63142/character-free-proof-that-frobenius-kernel-is-a-normal-subgroup/63156#63156 Comment by solovei solovei 2013-04-28T14:01:24Z 2013-04-28T14:01:24Z Showing it is a subgroup is a problem, but normality (invariant under conjugation) is easy. http://mathoverflow.net/questions/128918/why-didnt-finite-group-theorists-consider-groups-where-all-centralizers-of-non-i Comment by solovei solovei 2013-04-28T11:47:27Z 2013-04-28T11:47:27Z <a href="http://mathoverflow.net/howtoask#yourtitle" rel="nofollow">mathoverflow.net/howtoask#yourtitle</a> http://mathoverflow.net/questions/128918/why-didnt-finite-group-theorists-consider-groups-where-all-centralizers-of-non-i/128919#128919 Comment by solovei solovei 2013-04-27T14:39:57Z 2013-04-27T14:39:57Z @Geoff thanks. I just located the interesting page <a href="http://en.wikipedia.org/wiki/N-group_%28finite_group_theory%29" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> http://mathoverflow.net/questions/128841/classification-of-groups-in-which-the-centralizer-of-every-non-identity-element-i Comment by solovei solovei 2013-04-26T22:39:51Z 2013-04-26T22:39:51Z @Stefan Ok. But I have already asked a revised question for the class of amenable groups. http://mathoverflow.net/questions/128841/classification-of-groups-in-which-the-centralizer-of-every-non-identity-element-i Comment by solovei solovei 2013-04-26T19:24:55Z 2013-04-26T19:24:55Z Well the question is closed, although I don't know why. Was it too open-ended? should I have said characterize rather than classify? Would it have been better if I had restricted the class of groups? say to subgroups of GL_n(Z)? http://mathoverflow.net/questions/128841/classification-of-groups-in-which-the-centralizer-of-every-non-identity-element-i Comment by solovei solovei 2013-04-26T17:55:25Z 2013-04-26T17:55:25Z Andy, for finite groups I believe that groups where all centralizers are abelian can also be classified. http://mathoverflow.net/questions/128841/classification-of-groups-in-which-the-centralizer-of-every-non-identity-element-i Comment by solovei solovei 2013-04-26T17:28:25Z 2013-04-26T17:28:25Z that's a partial answer http://mathoverflow.net/questions/123460/cubic-fields-up-to-isomorphism/123507#123507 Comment by solovei solovei 2013-03-04T17:09:31Z 2013-03-04T17:09:31Z @Frank See page 77 of Elementary and Analytic Theory of Algebraic Numbers By Wladyslaw Narkiewicz. http://mathoverflow.net/questions/123460/cubic-fields-up-to-isomorphism/123507#123507 Comment by solovei solovei 2013-03-04T11:55:46Z 2013-03-04T11:55:46Z It is apparently known that the number of cubic fields with discriminant $D$ is unbounded, so by the bijection with the subgroups of the class group of $Q(\sqrt{D})$, that you mention, these subgroups are arbitrarily large. http://mathoverflow.net/questions/107902/finite-subgroups-of-sl-2r/107935#107935 Comment by solovei solovei 2012-09-26T00:44:29Z 2012-09-26T00:44:29Z So I think your argument might work over any field (of $char\ne 0$) so that the roots of unity in the field is just $\pm 1$. N'est pas? http://mathoverflow.net/questions/107902/finite-subgroups-of-sl-2r Comment by solovei solovei 2012-09-23T23:20:41Z 2012-09-23T23:20:41Z If the group is abelian then since every element is diagonalizable, the group is digonalizable over $C$ so the group is a subgroup of $C^*$. Now how to deal with the non-abelian case? http://mathoverflow.net/questions/107182/center-of-p-groups Comment by solovei solovei 2012-09-14T14:34:42Z 2012-09-14T14:34:42Z I should have written index $p^2$. http://mathoverflow.net/questions/106352/divisors-of-an-integer Comment by solovei solovei 2012-09-04T16:18:17Z 2012-09-04T16:18:17Z @quid exactly $M$ http://mathoverflow.net/questions/106352/divisors-of-an-integer Comment by solovei solovei 2012-09-04T15:14:18Z 2012-09-04T15:14:18Z I guess the growth rate is all you could expect to say anything about. http://mathoverflow.net/questions/106049/euclidean-inside-hyperbolic/106055#106055 Comment by solovei solovei 2012-08-31T21:14:24Z 2012-08-31T21:14:24Z on the hyperboloid