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Questions tagged [finite-groups]

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.

The questions that you can ask under this tag are of the same type that the questions related to the tag : homomorphisms of finite groups, representation theory, structure of finite groups...

You may refer to this Wikipedia entry for an introduction to this topic.

11774 questions
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G-invariants of a group

Let $\varphi:G\to GL(V)$ be a representation of a finite group $G$. Define the subspace $$V^G=\{v\in V\mid\text{for all } g\in G,gv=v\}.$$ How can I show that $V^G$ is a $G$-invariant subspace.
Sulayman
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homomorphismGroup for semidirect product

I am trying to construct in MAGMA the group $PSL(2, 81)⋊(Z_4 × Z_2)$ I can't make homomorphism from $(Z_4 × Z_2)$ to $Aut(PSL(2, 81))$. I tried the following G:=PSL(2,…
M.A
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What are various approaches of proving that a permutation group is a subgroup of another?

I have two permutation groups $G_1=\langle g_1,g_2\rangle$ and $G_2=\langle h_1,h_2\rangle$ acting on $\Gamma_1$ and $\Gamma_2$ respectively. I want to prove that $G_1 \leq G_2$. What are more nontrivial methods to do so?
Kaave
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$gH\subset Hg$ implies $gH=Hg$

Let $G$ be a finite group and $H
Cachorro
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Product of two elements of order q

Let $G$ be finite group. Let $x$ and $y$ be two elements of order a power of $q$, where $q$ is prime. Is the order of $xy$ equal to a power of $q$ (or of order 1)? thanks!
Jida
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Every Simple Abelian group is cyclic of prime order?

This was in a claim in my class notes in a proof that every Solvable Simple group is of prime order. I was able to verify it in the case where $G$ is finite, which I think might be a missing hypothesis in the statement of the theorem. Can anyone…
roo
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On a certain subset of a finite group of even order

Let $S$ be a subset of a finite abelian group of even order $G$ such that $x+y\not\in S$ for all $x,y\in S$, and the identity element $0\not\in S$. Define $-S=\{-x|~x\in S\}$ and $S-S=S+(-S)=\{x-y|~x,y\in S\}$. Suppose $S \cap (-S)=\varnothing$.…
LTX
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Size of two conjugate subgroups of a finite group

Given a finite group $G$ and any subgroup $H$. In all the examples I've looked at the intersection of two different conjugate subgroups of $H$ always had the same size. Is this always the case? I know that the intersection $H^{g_1} \cap H^{g_2}$…
Marc Bogaerts
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Set Theory/Cycles

Is there anyway/notation of taking a finite set lets say A={1,4,8,2} and turning it into cycle form where order matters for example?
W. G.
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An example of finite groups

Is there any example of finite group $G$ with the following properties? 1) There is prime divisor $p$ of order $G$ such that the number of cyclic subgroup of order $p$ is $p+1$. 2) The order of Sylow $p$-subgroup of $G$ is $p^{2}$. 3) $G$ is not…
N K
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Group action on finite set gives rise to a linear representation?

Let $G$ be a group that acts on a finite set $\Omega$. Let $V_{\Omega}$ be a $\mathbb{C}$-vector space whose basis elements $e_x$ are indexed by the elements $x$ of $\Omega$. In my book, it says that then we get a representation $\phi$ of $G$ on…
Roger
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Let $G$ be a finite group with cardinality not divisible by $3$. How can one show that for every $g \in G$ there is an $h \in G$ such that $g=h^3$?

Let $G$ be a finite group whose cardinality is not divisible by $3$. How can one show that for every $g$ belonging to $G$ there exists an $h \in G$ such that $g=h^3$?
user304330
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Question on the unsolvability a group

Let $G$ be a finite group. Let $\pi(G)=\{2,3,5\}$ be the set of prime divisors of its order. 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…
N K
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Meaning of Strong Primitivity

As J.D.Dixon noted in his great book; Permutations Group, we can speak about Strong Primitivity of a group acting on a set $\Omega$ by means of orbital graphs. The way he paved employes digraphs prove if such a group is strong primitive or not. I…
Mikasa
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If $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$

Question 1. Let $K\subseteq H\subseteq G$ and if $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$. 2. Let $K\subseteq H\subseteq G$ and if $K$ is a subgroup of $G$ and $H$ is a subgroup of $G$, then $K$ is a…
MTMA2
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