# 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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### What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such ...
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### Automorphism of a group is a group action [on hold]

Let G be a group and let $\Omega$ be a set. Then, the $Aut\left ( G \right )$ acts on $\Omega=G$ How can I show that this is true? Thank in advance.
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### An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
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### About decompositions of induced characters

Suppose $G$ is a finite group, $H\leqslant G$ is a subgroup. $\chi_1,...,\chi_s$ are all the irreducible characters of $G$ and $\psi$ is an irreducible character of $H$. Prove that if ...
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### $|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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### references of modular representations for finite group

What is modular representation for finite groups? I tried to find a book to understanding that but I could not find a good one. Are there any useful references?
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### Intersection of two subgroups with given information

suppose we know that $G$ is a finite group with order $43200$ and suppose that $H$ is a subgroup of $G$ with order $80$. Furthermore, assume that $K$ is also a subgroup of $G$ such that $[G:K]=1600$. ...
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### showing a set is a subgroup of a normaliser

Let $H$ be a subgroup of a group $G$ and defined $N_{G}\left ( H \right )=\left \{ g \in G \mid g^{-1}Hg=H \space\ \right \}$ Show that $H$ is a normal subgroup of $N_{G}\left ( H \right ).$ The ...
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### Specific condition for a map to be isomorphism

Let $G=\left ( \mathbb{R} \space\ \text{where} 0 \notin \mathbb{R},\cdot \right )$ and let r be a positive integer. Define $\phi:G\rightarrow G$ $x \mapsto x^{r}$ Show that $\phi$ is an ...
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### Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
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### Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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### Vertex-transitivity of the automorphism group of a digraph

I am trying to understand the theorem 3 of Cycles in graphs and groups by Kantor. Theorem $3$ If $G$ is a vertex-transitive group of automorphisms of a digraph $\Gamma$ with outdegree $d \ge 1$, ...
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### Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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### I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
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### Intuition of a theorem in Abstract groups

A theorem in Abstract groups Let $N \triangleleft G$ Then, $1\cdot$ If $H \leq G$ with $N \leq then H/N \leq G/N.$ Morever, if $N \leq K \leq G with K/N =H/N$ then ...
### $G$ has size $55$ and acts on a set of size $39$ [closed]
Let $G$ be a group with size $55$ which acts on a set with size $39$. Prove that this action has a fixed point (which means there exists a $x$ in the set such that $\forall g\in G : gx=x$.) I don't ...
Let G be a group and suppose $g \in G$. $\varphi:G\rightarrow Aut\left ( G \right )$ $g \mapsto i_{g}$ is a Homomorphism with image $Inn\left ( G \right )$ where \$Inn\left ( G \right )=\left \{ ...