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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If $p\mid|G|$ then how many elements of order $p$ are there in $G$?

Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ ...
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The comultiplication on $\mathbb{C} Q$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
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How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $\ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
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Show that there is just one subgroup $H \subset S_4$ such that $[S_4:H] = 2$

I had to show that $S_4$ has one subgroup of index $2$. Below, you'll find what I tried to do so Of course, $A_4$ is a subgroup index $2$. To show that there is not another subgroup with the same ...
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how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...