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.
12,281 questions
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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 $$\mathrm{Aut}(B_i)=\{g\in\mathrm{GL}_n(\mathbb{Z})\mid\ gB_ig^{\...
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The consequences of the orbit-stabilizer theorem
Let G be a group acting on a set A. Let $[x]$ denote the orbit of any $x\in A$. Also let $G_x$ denote the stabilizer of $x$.
From the orbit-stabilizer theorem, the orbit of any $x\in A$ has the same ...
- 6,188
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A Consequence from Sylow Theorems on Conjugacy of all $p$-Sylow groups
Let $G$ a finite group, $p$ a prime number, $P$ a non trivial $p$-Sylow group of $G$ (i.e., $\vert P \vert =p^n$ with $n \ge 1$ for $\vert G \vert =p^nm$ with $(p,m)=1$) and $Q \leq G$ any $p$-group. ...
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Question about transitive conjugation actions
Let $A, B$ be finite groups. Suppose $A \triangleleft B$ with $B$ transitively acting upon $A \setminus \{1\}$ by conjugation. (This implies $A \cong (\mathbb{F}_p^n, +)$.) Must there exist $C$ with $...
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Conditions on $F(C)=F(G)$ for Bender's theorem
https://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/finitegroups2010.pdf
As shown in page 48 of Theorem 3.30 in above link. In the study of generalized Fitting subgroup of $G$. Let $E(G)$ be the layer ...
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Action on the second group of cohomology
Let $G$ be a finite group and let $M$ be a G-module. So we are given an action
$$G\times M\to M$$ by $(g,m)\mapsto g.m$.
Let $H^2(G,M)$ be the second cohomolgy group. Define the following action of $G$...
- 95
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Hecke algebras over fields other than $\mathbb{C}$
I am trying to prove some statements about specific Gelfand pairs, when considering representations of some finite groups over field $k$ which can be of positive characteristic. For this I study some ...
- 994
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Orbits of a cyclic group on words of length $L$ with alphabet of size $k$ [duplicate]
Consider an alphabet $\Sigma$ of size $k=|\Sigma|$ and the set of words $\Sigma^L$ of length $L$, with the natural action of the cyclic group $\mathbb{Z}_L$ with generator $\tau$ acting naturally as $\...
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problems in Group Theory [duplicate]
I am looking for a good sources for practice problems in Abstract algebra I ( group theory) starting from beginner to harder problems. Any idea
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Proving $\sum_{\substack{G \ p\text{-abelian} \\ \#G \leq p^n}}\frac{1}{\#\operatorname{Aut}G}=\prod_{i=1}^n(1-p^{-i})^{-1}$
The problem involves proving the following identity:
$$
\sum_{\substack{G \ p\text{-abelian} \\ \#G \leq p^n}} \frac{1}{\#\operatorname{Aut}G} = \prod_{i=1}^n (1 - p^{-i})^{-1}
$$
As $ n $ approaches ...
- 171
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Wang-Chen theorem on solvability?
There is a theorem by Wang and Chen that says: when the finite group $A$ acts via automorphisms on the finite group $G$ with $|A|$ and $|G|$ coprime, and $C_G(A)$ is either odd-order or nilpotent, ...
- 6,050
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Finite group with $H^3(G,U(1))$ trivial
Let $ G $ be a finite group. Today I learned that, for a certain category made from $ G $, the different associator structures making it a tensor category correspond to 3-cocycles, and the number of ...
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Proof of theorem of Burnside ( Anyone who have the Steinberg's Representation theory of finite groups book ? )
I am reading the proof of Theorem 9.1.10 in the Benjamiin Steinberg's book, representation theory of finite groups and stuck at some statement.
First let me arrange associated preliminaries
...
- 4,056
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Is the set of elements commuting with *some* power of every element always a subgroup
Let $G$ be a finite group. For $y\in G$ of order $n$, say that $x\in G$ $n$-permutes with $y$ if there exists some $1 \leq i < n$ such that
$$
xy^i = y^i x.
$$
Define
$$
T := \{x \in G : \forall\, ...
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Exponent of an abelian commutator subgroup $G'$ divides the exponent of $G/G'$
Let $G$ be a finite group. I want to prove that the exponent of an abelian commutator subgroup $G'$ divides the exponent of $G/G'$. The exponent of a group is the smallest positive integer $n$ such ...
- 125
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Show a group of order $2^{11}\left(2^{11} - 1\right)$ is not simple.
I was able to show that either a Sylow $89$-subgroup is normal or its normaliser is of order $2047$, and similarly for the Sylow $23$-subgroup, but then got stuck.
- 191
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About the hypercenter of a group
The hypercenter of a group is defined by $Z_{\infty}(G)=\bigcup\limits_{i} Z_{i}(G)$. My question is
if $H$ is a nilpotent subgroup of $G$, then $HZ_{\infty}(G)$ is nilpotent.
How to prove that $Z_{\...
- 111
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(Un)equivalence of two definitions "$R$-sequencing" in group theory
I have a question considering two different definitions of "$R$-sequencing".
If we take a look at the M. A. Ollis' article "Constructing R-sequencings and terraces for groups of even ...
- 143
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A transitive subgroup of order $n$ of $S_n$ inducing a non-associative latin square.
For $n\in\mathbb N$, the multiplication table of a quasigroup $Q=([n],*)$ induces a transitive subset $\Sigma=\{\sigma_1,\dots,\sigma_n\}\subseteq S_n$ via the position:
$$\sigma_i(j):=i*j\tag1$$
and ...
- 5,581
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Geometric representation of finite groups
Looking for confirmation that the following method constructs a geometric object whose symmetry is described by a finite group $G$.
Let $G$ be a finite group which is a subgroup of the symmetric group ...
- 556
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A question about the hypercenter of a finite group
The background of my question is from https://www.sciencedirect.com/science/article/pii/0021869377902927.
Lemma 1
Let $G$ be a finite group. A normal $p$-subgroup $N$ of $G$ lies in $Z_{\infty}(G)$ if ...
- 111
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How to guarantee the completeness of classification of finite group with certain properties?
I was looking at classification of finite groups with certain properties(i.e, group with abelian automorphism group) recently. During this process, I feel quiet confusing that when can we say we "...
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Whether the kernels of projections of a compact profinite group are open?
This is Exercise 23 in Section 17.2 in Dummit&Foote's Abstract Algebra:
Suppose $G$ is a compact topological group. Prove the following are equivalent:
(i) $G$ is profinite, i.e., $\displaystyle ...
- 763
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Transitivity check runtime
A necessary condition for a subset $\Sigma\subseteq S_n$ to be a transitive permutation group of order $n$, is to be... transitive. Is the best algorithm to check $\Sigma$'s transitivity faster than ...
- 5,581
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Information required to determine a group given a minimal generating set
Say we have a finite group $G$ generated by $\langle g_1,g_2,\ldots ,g_n\rangle$ which is a minimal generating set. What information is required from this set in order to determine the whole group?
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