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.
12,285 questions
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Cyclicity of the multiplicative group of the integers modulo a prime
Edit. It seems that the Lemma 2 needs already the existence of a primitive root modulo $p$. If there's no other way to prove it, then my argument is pointless.
(NB: I'm aware that there's plenty of ...
- 5,601
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Automorphism group of $UT(3,p)$
Here $p>2$ is a prime, the group $UT(3,p)$ is the group of $3\times3$ upper unitriangular matrices with coefficients in $\mathbb{F}_p$:
$$\begin{bmatrix}
1&x&y\\
0&1&z\\
0&0&...
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Center of a finite perfect group
This question is distantly related to the following MathStack post: How "big" can the center of a finite perfect group be?
The above post and its answers comment on the size of the center of ...
- 917
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Automorphisms of integral quadratic forms
For $i=1,2$, let $B_i\in\mathrm{SL}_n(\mathbb{Z})$ be two symmetric positive definite matrices.
We define their automorphism groups as $$\mathrm{Aut}(B_i)=\{g\in\mathrm{GL}_n(\mathbb{Z})\mid\ gB_ig^{\...
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In a semidirect product $NK$, can GAP find the subset of $Aut(N)$ induced by the elements of $K$?
Let $G$ = $NK$ be a semidirect product of N and K. Let $\sigma$ be the mapping from $G$ to $Aut(N)$ given by
$\sigma(g)$ = $gng^{-1}$. I'm trying to use GAP to find $\sigma$ and $\sigma(K)$. Is ...
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3
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1
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The consequences of the orbit-stabilizer theorem
Let G be a group acting on a set A. Let $[x]$ denote the orbit of any $x\in A$. Also let $G_x$ denote the stabilizer of $x$.
From the orbit-stabilizer theorem, the orbit of any $x\in A$ has the same ...
- 6,197
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Minimal finite non-solvable groups whose order has exactly three distinct prime factors.
Let $G$ be a finite group. Define $\pi(G)$ to be the number of distinct prime factors of $|G|$. It is known that any finite group $G$ with $\pi(G)\leq 2$ is solvable. Also there exists many non-...
- 917
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A Consequence from Sylow Theorems on Conjugacy of all $p$-Sylow groups
Let $G$ a finite group, $p$ a prime number, $P$ a non trivial $p$-Sylow group of $G$ (i.e., $\vert P \vert =p^n$ with $n \ge 1$ for $\vert G \vert =p^nm$ with $(p,m)=1$) and $Q \leq G$ any $p$-group. ...
- 10.1k
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Decomposition of finite-order element into commuting components of coprime order
This was a question posed at the end of a problem sheet in a group theory class I am taking:
Problem
Let $G$ be any group, let $g$ be any element of $G$ of finite order, and let $p$ be any prime.
Show ...
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Question about transitive conjugation actions
Let $A, B$ be finite groups. Suppose $A \triangleleft B$ with $B$ transitively acting upon $A \setminus \{1\}$ by conjugation. (This implies $A \cong (\mathbb{F}_p^n, +)$.) Must there exist $C$ with $...
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Consequence from Sylow Theorems on Conjugacy of all $p$-Sylow groups
Let $G$ a finite group, $p$ a prime number, $P$ a non trivial $p$-Sylow group of $G$ (i.e., $\vert P \vert =p^n$ with $n \ge 1$ for $\vert G \vert =p^nm$ with $(p,m)=1$) and $Q \leq G$ any $p$-group. ...
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Conditions on $F(C)=F(G)$ for Bender's theorem
https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/finitegroups2010.pdf
As shown in page 48 of Theorem 3.30 in above link. In the study of generalized Fitting subgroup of $G$. Let $E(G)$ be the layer ...
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Generating a finite non-solvable group with an element and its conjugate
It is known that for any finite simple group $G$ there exist two elements $a,b\in G$ such that $a$ and $b$ are conjugates in $G$ and $\langle a, b \rangle=G$. My question: Is it true for any finite ...
- 917
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Action on the second group of cohomology
Let $G$ be a finite group and let $M$ be a G-module. So we are given an action
$$G\times M\to M$$ by $(g,m)\mapsto g.m$.
Let $H^2(G,M)$ be the second cohomolgy group. Define the following action of $G$...
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What's the minimum maximum index of a subgroup chain connecting the symmetric group to the trivial group?
Given the symmetric group $S_n$, consider a chain of subgroups
$$S_n=G_k>G_{k-1}>\cdots>G_2>G_1>G_0=1$$
and let
$$m=\max_{1\leq j\leq k}\frac{|G_j|}{|G_{j-1}|}.$$
Define $\chi(S_n)$ as ...
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