# 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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### Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
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I am having trouble with this problem: Assume that $(\mathbb{G}, *)$ is a finite group and there exists a positive integer $n$ such that gcd($n, |\mathbb{G}|)=1$. Prove that the function $F_n: \... 1answer 113 views ### Finite group$G$is product of a subgroup$H$and normalizer of a Sylow$p$-subgroup of$H$Let$G$be a finite group,$H$a normal subgroup of$G$and$P$a Sylow$p$-subgroup of$H$. Let$N_G(P)$be the normalizer of$P$in$G$. Show that$G=N_G(P)H$. 1answer 96 views ### Sylow subgroups of soluble groups Suppose$G \leqslant S_p$acts transitively on$\{1,...,p\}$for prime$p$. Let$P \leqslant G$be a Sylow p-subgroup. Is it true that$G$is soluble <=>$P \triangleleft G$? 1answer 155 views ### Elements of order 3 in$PGL(4,\mathbb{R})$I need to classify all elements of order 3 up to conjugation in$PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead$GL(4,\mathbb{C})$... 0answers 149 views ### Central extensions of elementary abelian p-groups Given an elementary abelian$p$-group$E$, we can consider$E$as a trivial$E$-module; my first question is : How can one compute the rank of of the cohomology group$\operatorname{H}^n(E,E)$,$n \...
I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...