User niki - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:00:01Z http://mathoverflow.net/feeds/user/24501 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102908/the-number-of-cyclic-groups-of-order-2p-in-a-group The number of cyclic groups of order 2p in a group Niki 2012-07-23T04:21:34Z 2012-07-23T11:36:47Z <p>Let $t_{k}$ be the number of elements of order $k$ in the group $G$. It known that if $|P|=p$ (where $P$ is Sylow $p$-subgroup of $G$), then $t_{2p}$ is a multiple of $t_{p}$. Now let $|P|=p^{2}$, the number of Sylow $p-$subgroups be $1$ and the number of cyclic subgroups of order $p$ be $p+1$. Also let the number of cyclic subgroups of order $2$ be $p(p+1)/2$. Is it true the number of cyclic subgroups of order $2p$ is a multiple of $p+1$? Or: Is it true $t_{2p}$ is a multiple of $t_{p}$? Thanks in advance.</p> http://mathoverflow.net/questions/99824/why-a-group-of-order-2m-cdot-pn-cdot-qt-is-solvable Why a group of order $2^{m}\cdot p^{n}\cdot q^{t}$ is solvable? Niki 2012-06-17T07:57:47Z 2012-06-18T11:21:34Z <p>It's known all groups of order $p^{m}q^{n}$ and all groups of odd order are solvable (By Burnside theorem and Feit-Thompson theorem).</p> <p>Let $G$ be a group of order $2^{m}\cdot p^{n}\cdot q^{t}$ where $p\neq 3$ and $q\neq 3$ are prime.</p> <p>Does anyone knows why $G$ is solvable?</p> http://mathoverflow.net/questions/99780/question-on-the-unsolvability-of-a-group Question on the unsolvability of a group Niki 2012-06-16T11:46:33Z 2012-06-16T11:53:11Z <p>Let $G$ be a finite group. Let $\pi(G)={2,3,5}$ be the set of prime divisors of its order of $G$. If 6 divide the number of Sylow 5-subgroups of $G$ and 10 divide the number of Sylow 3-subgroups of $G$, then whether the group $G$ group with those properties is unsolvable? Thank you so much.</p> http://mathoverflow.net/questions/102908/the-number-of-cyclic-groups-of-order-2p-in-a-group Comment by Niki Niki 2012-07-23T11:44:26Z 2012-07-23T11:44:26Z @Derek Holt: My question asked 7 hours ago and you have given a counterexample 2 hours ago on Math StackExchange. http://mathoverflow.net/questions/99780/question-on-the-unsolvability-of-a-group Comment by Niki Niki 2012-06-18T04:30:48Z 2012-06-18T04:30:48Z @Anthony Quas: I asked my question in <a href="http://math.stackexchange.com" rel="nofollow">math.stackexchange.com</a> and got a answer you can see it. http://mathoverflow.net/questions/99824/why-a-group-of-order-2m-cdot-pn-cdot-qt-is-solvable/99829#99829 Comment by Niki Niki 2012-06-17T09:16:15Z 2012-06-17T09:16:15Z @Rahim khan, @Mark Spair:Thank you Rahim and Mark, very nice proof. http://mathoverflow.net/questions/99780/question-on-the-unsolvability-of-a-group Comment by Niki Niki 2012-06-17T03:37:03Z 2012-06-17T03:37:03Z @Yemon Choi: That's right in the special case it is easy but, in the general case this is not easy question. http://mathoverflow.net/questions/99780/question-on-the-unsolvability-of-a-group Comment by Niki Niki 2012-06-16T19:05:56Z 2012-06-16T19:05:56Z @Anthony Quas: I'm sure you do not know what the answer. http://mathoverflow.net/questions/99780/question-on-the-unsolvability-of-a-group Comment by Niki Niki 2012-06-16T13:36:08Z 2012-06-16T13:36:08Z In particular if the number of Sylow $5$-subgroups of $G$ is 6 and the number of Sylow $3$-subgroups of $G$ is 10, then by the Hall's theorem $G$ is unsolvable group. Because if $G$ is solvable, then $2\equiv 1$ (mod $5$), a contradiction.