# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### $G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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### Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
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I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ... 2answers 33 views ### Questions about Sylow$p$-groups Question 1 Is it true that there is only one Sylow$p$-group in an abelian group? Question 2 If there is only one Sylow$p$-group, then it is normal, true? Because if$H\leq G$is a Sylow$p$-group ... 2answers 25 views ### Trying to find an isomorphism I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ... 5answers 508 views ### A criterion for a group to be abelian [duplicate] I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let$m$... 1answer 36 views ### Sylow$p$-groups in$GL_3(\Bbb F_p)$Fix a prime$p$. How many Sylow$p$-groups are there in$GL_3(\Bbb F_p)$? Let$s_p$be the number of Sylow$p$-groups in$GL_3(\Bbb F_p)$. By the Sylow theorems,$s_p\equiv 1\mod p$and if$p^e$... 1answer 41 views ### Sigma and Pi Chemistry/Math Permutation Question Does anyone know if sigma and pi bonds in chemistry have any mathematical definition? The reason I'm asking this is because I've recently read a lot about cycles and permutations, and they seem to ... 2answers 49 views ### Can the order of$x \in U_{31}$be$10$? Does there exist an element$x\in U_{31}$such that the order of$x$is$10$? Here$U_{31}$is the group of units of$\mathbb{Z}/31\mathbb{Z}$. 3answers 33 views ### Unique order$d$subgroup of$\Bbb Z/n\Bbb Z$if$d\mid n$[duplicate] Let$G=\Bbb Z/n \Bbb Z$and assume$d\mid n$. There exists a unique subgroup of$G$of order$d$. The mod$n$reduction of$n/d\in \Bbb Z$generates a$d$element subgroup of$\Bbb Z/n\Bbb Z$, ... 0answers 39 views ### Motivation of indices of all subgroups of symmetric group$S_n$In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group$S_n$acting on$n$objects ? Three submissions was submitted in 1860, no ... 1answer 26 views ### Cohomology ring of$G$based on its Sylow. I have a bunch of notes made from a professor about cohomology that states that If$S$is a$p$-Sylow subgroup of$G$($\vert G \vert <\infty$), then $$H^{\ast}(G,\mathbb{F}_p)\leq H^{\ast}(... 1answer 53 views ### Need help in understanding the example from Dummit & Foote text This is an example from Dummit & Foote text. Let D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>.Since [r,s]=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...