Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Automorphism group of a scheme, 2
Hi,
I have the following two questions about automorphism groups of schemes.
First of all, let $S$ be a scheme, and $S^c$ its set of closed points. What
is the connection between $Aut(S)$ and $Aut(...
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Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
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Classes of groups with finitely many retracts
Let $G$ be a group. A subgroup $H$ of $G$ is called a retract of $G$ if there exists a homomorphism $r:G \to H$ such that $r(h)=h$, for all $h\in H$.
We can easily check that every finite-by-cyclic ...
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A particular permutation on $\mathbb{Z}_n$
Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$.
Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying :
$$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
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Number of reduced decompositions of the dihedral group $D_6$ [closed]
The Weyl group of $\frak{g}_2$ is the dihedral $D_6$. Let us denote its longest element by $w_0$. How many reduced decompositions does $w_0$ have?
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Fourier transform on lattice strip
I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
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Sylow $p$-subgroups of FSym($\mathbb N$)
$\DeclareMathOperator\FSym{FSym}$Let $\FSym(\mathbb{N})$ denote the finitary symmetric group on the set of natural numbers. How many Sylow $p$-subgroups does $\FSym(\mathbb{N})$ have for any prime $p$?...
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Given the index of two permutations, Is there a direct way to compute the index of their composition? [closed]
Suppose you are given two indexed permutations, (7 followed by 4, for instance)
What is the best way to go about finding their composition, given the indices themselves? I'd imagine that the answer ...
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measures on groups without assuming a locally compact group topology
I'm interested in knowing whether there exists any kind of theory for measures on groups without assuming that it's the Haar measure for a locally compact group topology.
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A question on Frobenius groups [closed]
Please change the title if needed.
Let $p$ and $q$ be distinct primes and $G\cong(\underbrace{\mathbb{Z}_{q}\times\mathbb{Z}_{q}\times\dots\times\mathbb{Z}_{q}}_{n\,\,times})\rtimes\mathbb{Z}_{p}$, ...
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About generator of minimal length coset representatives
Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W_J = \langle s: s \in J\rangle \subseteq W$.
Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$.
...
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Automorphisms of nilpotent groups of class two
Is there any article that help me study automorphisms of nilpotent groups of class two with cyclic center?
In "Odd order nilpotent groups of class two with cyclic centre, Y. K. Leong (1974)" there is ...
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How do we prove that the following implication in semiring? [closed]
Let $G$ be a group. Clearly the power set $(\mathcal{P}(G),\cup,. )$ is the semiring, where $\cup$ means ordinary union and '.' is defined as $$AB= \left\lbrace ab \in G \mid a\in A\mbox{ and } b\in B ...
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Reference request: Any connected Lie group has a countable base for its topology
I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?
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Orthogonal idempotents with sum equal to 1 in $k[G]$ span sub-Hopf algebra
Let $G$ be a finite group. Let $B$ be a set of orthogonal non-zero idempotents with $|B| \leq |G|$, s.t. $\sum_{b \in B}b =1_{kG}$. Is it known if $B$ spans a sub-Hopf algebra $kH \subseteq kG$?
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Minimum number of generators of a finite group
Let N be a cyclic normal subgroup of a finite group G and $\frac{G}{N}$ be a
p-group in which p and the order of N is coprime. Then what we can say about the relation between the minimum number of ...
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Difference between $G$-rank and the maximal $G$-power quotient
Let $G$ be a finite simple group, and $F$ a profinite group (I'm really interested in the case where $F$ is free of finite rank, in particular rank 2).
In Ribes-Zalesskii, they define the $G$-rank of ...
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p-groups with special property on its centralizers
Thanks for any help or comment.
Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...
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Relation between the Frattini property and pronormal subgroups of solvable groups
A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N_L(H)K$.
A subgroup is $H$ is pronormal in $G$ if for ...
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Uniform pro-p groups as a semi-direct product
Let $G$ a finitely generated uniform pro-$p$ group. Then $G/[G,G]$ is abelian and so it is of the form $\mathbb{Z}_p^r\times T$ for some integer $r$ and finite $p$-group $T$. Therefore, $[G,G]$ is ...
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Which finite groups can be characterized by their automorphism groups?
Given a finite group G, we denote by Aut(G) the group automorphisms of G . Which finite groups G can be characterized by the group Aut(G), i.e. Aut(H)≅Aut(G) implies H≅G?
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Characterizing cyclic group of order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, by Lattice isomorphisms
Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any group.Assume that $L(G)≅L(H)$and Aut$(...
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Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?
Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...
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Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter
Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 (...
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The name of a group of order 24 [closed]
I encountered a group $G =\langle(1,3,2,4),(3,5,4,6)\rangle\subseteq S_6$ in my study, but I do not know its name.
Let $f=(1,3,2,4)$ and $g=(3,5,4,6)$. We have $g^2=fg^2f$, and thus $\langle f,g^2\...
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direct product of a finite group with an infinite symmetric group [closed]
Cross-posted from MSE: https://math.stackexchange.com/q/1226622/15624.
Let $G$ be any finite group, and $S_{\aleph_0}$ the group of all bijections $\mathbb{Z}\rightarrow \mathbb{Z}$.
Is $G\times S_{\...
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A problem with pointwise stabilizer subgroups of fixed-point subspaces I
Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...
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The simple groups with an absolutely irreducible projective representations with small degrees
In the question "Simple Hurwitz Groups of order less than 10^7", it came up that all the Hurwitz groups with absolutely irreducible projective representations of degrees up to 7 over any field are ...
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Possible degrees of faithful projective representations of $\mathrm{PSL}(k,q)$ and $\mathrm{Sp}(2k,q)$ over complex numbers
Let $q$ be a prime power and $k$ a positive integer. What are the possible degrees of faithful projective representations of the projective special linear group $\mathrm{PSL}(k,q)$ (over the Galois ...
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Amenability of the Koopman representation
Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi_X:G\rightarrow U(L^2(X,\mu))$ defined by
$(\pi_X(g)\xi)(...
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Is a weakly separable group always Lindelöf?
By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
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Centralizers of p-elements in general linear groups
Let $G=GL(n,q)$ be a generalized linear group whose center is a cyclic group of prime order $p$. Does there always exists an element $x\in G$ such that $C_G(x)$ is a group of exponent $p$?
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What are these subgroups called? [closed]
Let $G$ be a group. Let $S \subset G$. Consider the set of all $x \in G$ such that $xS = S$.
What is this unique largest subgroup of $G$ preserving $S$ under left-multiplication called?
(As for the ...
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when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
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Accessible problems on classical groups over complex or real numbers.
I am a undergraduate student doing project with my professor in group theory. I am Looking for some accessible problems for undergraduate on Classical groups over complex or real numbers (particularly ...
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Classification for a special simple group
The prime graph of finite group $G$, is as follows: the vertex set is prime
divisor of $|G|$ and two distinct vertices $p$ and $q$ are joined by an edge
if and only if $G$ has an element of order $pq$....
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$p$-primary then divisible?
I asked this via MathSE, but haven't got any responces. Sorry for asking it here. Sorry.
We know that in the context of abelian groups, $p$-groups are called $p$-primary groups. I have a question ...
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For an algebraic group acting on a variety, why are orbits representable?
I suspect this is really obvious, but I'm not seeing it.
For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be $G(...
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Automorphism Group of a p-group (finitely generated)
Does someone know whether the order of the automorphism group of a general p-group of order $p^n$ is bounded from above by $(p^n)^2 $? (Every element can possibly be transferred to one of other $p^n$ ...
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Can we have a finite cyclic group of rational numbers (under multiplication)? [closed]
I am a PDE guy, who works in imaging. We are trying to exploit the inherent group structure
of an image, i.e. we consider a representation of the similitude group ($SIM(2)=\mathbb{R}^2\ltimes(SO(2)\...
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Non-elementary examples of nearly normal subgroups
$H$ is called a nearly normal subgroup of a group $G$ if it is of finite index in its normal closure, $H^G:=\cup_{g\in G} gHg^{-1}$, in $G$. Clearly, every normal subgroup or subgroup of finite index ...
user23860
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List of small dimension Lie group irreps
For semisimple Lie groups it's just a lookup (I define "small" as dim(R)<5 and
put the dimensions of the Clebsch-Gordon series of RxR in parentheses):
A1(1*1=1),A1(2*2=1+3),A1(3*3=1+3+5),A1(4*4=1+3+...
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About 5.4, page 54 of J.-P. Serre's Book "Trees"
Definitions:
Let $G$ be a group wich acts without inversion on a connected nonempty graph $X$. We shall see that, if $X$ is a tree, then $G$ can be identified with the fundamental group of a certain ...
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homeomorphism of topological group
Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ ...
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If H is a subgroup of Z(G) and G/H is nilpotent, then G is nilpotent. [closed]
Can you help me and give me the proof of this statement please? And can you explain me why this statement is not true when $H$ is not a subgroup of $Z(G)$? Thank you very much
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Is a group a direct product? [closed]
Given a group G
How can one tell if it could have been formed as the direct product of sub groups.
If so what are the groups.
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groups and asymmetry [closed]
In an informal sense, groups are related to symmetry. I was wondering if there are groups that describe some sort of asymmetry. Does anyone know of such groups or is asymmetry and groups a ...
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A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
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Combinatorial problem in $G(54, \, 5)$ - Reprise
This post is a follow-up to my previous post MO479127. I am trying to concentrate on a subset of relations, hoping to find some structure on the set of solutions that explains why the whole set of ...
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Hyperoctahedral group, preliminaries [closed]
I am looking for information on the hyperoctahedral group
From what I understand, the hyperoctahedral group is 'the generalized symmetric group' in the case where $m=2$. That is, the hyperoctahedral ...
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