Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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From direct sum of quotient group of a group to direct sum of the group
We know that for a cyclic group $G$, if $G=A\oplus B$, then for some subgroups $H$ of $G$, We have $G/H=(A+H)/H\oplus (B+H)/H.$ But, if we know that for a subgroup $H$ of $G$, $G/H=(A+H)/H\oplus (B+H)...
- 21
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Why does this proof on the cyclicity of a prime multiplicative group not conclude that the solutions to a polynomial biject the powers of one element?
This argument comes from the first proof in Keith Conrad's collection of proofs that multiplicative groups of prime-order cyclic groups contain at least one generator.
The proof asks the reader to ...
- 101
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134
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How can Borel-de Siebenthal theory be generalized?
Borel-de Siebenthal theory can be thought of as an algorithm that, given a semisimple compact Lie group $G$, gives all semisimple compact Lie subgroups whose root systems have the same rank as $G$’s.
...
- 2,773
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215
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Subgroups generated by two random elements
Suppose that we have a finite group $G$ and choose elements $a, b \in G$ at random. What can be said about the order of the subgroup generated by $a$ and $b$? Mainly, what is the expected order, $\...
- 2,773
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102
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On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k
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151
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zero divisors of group ring when the group is abelian
Let G be an abelian group with torsion and C[G] be the group ring over complex numbers C. Is there a clear description or classification of zero divisors of C[G]?
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154
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Coxeter groups and finitness of number of roots
Take any graph $\Gamma$ with $n$ vertices $\{v_1, v_2 \dots v_n\}$, and associate to this graph it's set of simple roots i.e. a vectors of the canonical basis $e_i, \ i=1..n$ of $R^n$ for each vertex $...
- 131
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121
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General linear group analogs
The Wikipedia pages for $E_6$ and $E_7$ list three series of groups notated as each of $E_6(q)$, $^2E_6(q)$, and $E_7(q)$:
The simple form, analogous to $\operatorname{PSL}_n(q)$
The adjoint form, ...
- 2,773
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114
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The action of an extension group $G=p^{1+2n}{.}Q$ on the faithful characters of its normal subgroup $p^{1+2n}$
Let $G=p^{1+2n}{.}Q$, $n>1$, be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the ...
- 49
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77
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Describing the ordinary irreducible characters of a special $p$-group $p^{n+m}$
Let $P$ be a special $p$-group $p^{n+m}$. So $P$ will have $p^m$ linear characters. How does one describe (or determine) the other ordinary irreducible characters of $P$ and will they all be ...
- 49
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67
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Factor group of direct product by restricted direct product 2
Let $W:=\prod_{i\in \omega} F_i$ be the (external) unrestricted direct product and $U:=\prod_{i\in \omega}^w F_i$ be the (external) restricted direct product of finite groups $F_i$ such that $\lvert ...
- 69
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Another question concerning finite metacyclic groups
Given a non-split finite metacyclic group $H$, does there always exist a finite split metacyclic group $G$ with a normal cyclic subgroup $N$ of prime power order such that $H \cong G/N$?
Based on my ...
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301
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Inverse Galois problem on simple groups
Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group.
I've tryied to mess with the embedding problem for ...
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169
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A group acts on a groupoid
Let $G$ be a group. Let $(\Pi,\circ)$ be a groupoid. Suppose I have a $G$-action on every morphism space $\Pi(p,q)$, denoted by $G\times \Pi(p,q)\to \Pi(p,q)$, $(g, \sigma)\mapsto g\cdot \sigma$. (For ...
- 2,789
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Large subgroups of infinite-dimensional vector spaces
Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$.
Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
- 4,547
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150
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How to classify rings by combinatorial structures?
There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
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274
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Algorithm to compute automorphism group of a finite group
Is there an algorithm to compute automorphism group of a finite group?
GAP has a function to do this, but while perusing their GitHub repo, I could not find an implementation. I'm struggling to find ...
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152
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Left-side cosets of an open subgroup
Let $G$ be a topological group and $H$ its closed subgroup. $K$ and $L$ are open subgroups of $G$ and $H$ respectively. Let $g_{1}, g_{2}\in G$. We assume $L\cap g_{1}K\neq \emptyset$ and $L\cap g_{2}...
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Is a transfer homomorphism surjective?
Let $G$ be finite group with minimal number of generators$d$, and all his proper subgroups have at most $d-1$ as minimal number of generators.
Fix a normal subgroup $N$ of $G$.
For all subgroups $H$ ...
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134
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Intersection of descending series in a free group
I have stumbled upon a problem. It can be stated in the following way: Let $E$ be a finitely generated free group. Denote $\gamma_n(E)$ the $n$-th term of the lower central series. Consider a ...
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102
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classification for some groups
Let $G$ be a finite group. Suppose that $G$ acts on a set, say $X$, transitively such that for every $x\in X$, $G_x^g=G_x$ or $G_x^g\cap G_x=\{1\}$. Could you please tell me if there is a ...
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185
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Isomorphic Coxeter groups
After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
- 2,773
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181
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Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"
I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".
I have only (not yet enough!) standard background on the ...
- 1,013
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167
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Is there a theory of "partial" group actions?
I am looking for references that may formalize the following idea: let $R = k[X]$ be the coordinate ring for a generic $n \times m$ matrix $M$. It is well known that the ideal of $r \times r$ minors ...
- 553
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112
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Subgroup isomorphism problem
I was searching for the graph isomorphism problem because I was looking for NP-COMPLETE problems to write about in a article I`m writing.
Learning about the problem a thing came to my mind, does this ...
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250
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Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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73
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Mapping property of $p$-Sylow groups of profinite groups
Let $G$ be an abelian profinite groups. Then we have the Sylow group decomposition
$$G\cong \prod_p G_p.$$
In the case of finite groups, we have $ \prod_p G_p\cong \bigoplus_p G_p$ and thus
$$\text{...
- 4,418
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99
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Unimodular matrices fixing $(1, 1, \cdots, 1)$
What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
- 356
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Help for literature on entrywise invariant kernels
I am looking for literature on entrywise invariant kernels.
The specific example I have in mind is $K:R^{d}\times R^{d}\to R$ and locally compact groups acting on vector space $R^{d}$.
More precisely ...
- 329
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131
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Very transitive groups
For a project for one of our subjects we have to write about a certain topic.
Our topic is very transitive groups. This means a permutation group on the natural integers, that is $k$-transitive for
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74
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Estimate of the nilpotency class from the subgroup
Let $G$ be a nilpotent group and $H \vartriangleleft G$ a normal subgroup such that $[G:H] \le m$.
Assume $H$ has the nilpotency class $ \le n$. Can we show the nilpotency class of $G$ is bounded by a ...
- 2,535
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101
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sum of all subgroup elements
1.start from multiplicative group modulus N where N is odd.
2.take all elements of subgroup with generator equal 2.
question : what do you need to know about N
(factorization,phi) for fast ...
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When is the natural map of Tate cohomology an isomorphism?
First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate cohomology groups which in ...
- 1,215
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82
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Fast double exponentiation in finite fields
Let $p$ be a prime, and let $\mathbb{F}_p$ be the finite field with $p$ elements. Let $a$ be a non-zero element of $\mathbb{F}_p$. Can we quickly evaluate $a^{2^r} \mod{p}$? Using repeated squaring, ...
- 1,703
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81
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Semicharacters on groups
I think, this must be simple, but apparently, I don't have enough intuition, so excuse me. The following construction is useful in the holomorphic duality theory for complex Lie Groups:
Let $G$ be an ...
- 7,426
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141
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Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 356
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149
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Upper-triangular matrices as union of centralizers of cyclic elements
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The ...
- 463
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Making the link relationships of a subdivided icosahedron symmetric
Consider the vertices $v_i$ of a subdivided icosahedron $J$. In my case, each vertex $v_i$ has an ordered tuple of nodes denoting the edges of $J$ starting in $v_i$. All vertices have 6 edges, except ...
- 101
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79
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Large gaps in the norm of a subgroup and its centraliser
Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$.
Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$,
...
- 4,432
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81
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QI-closure of $\mathrm{NA}\times\mathrm{NA}$
Suppose we know the following about a class of groups $\mathcal{G}$.
If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$.
If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is ...
- 6,652
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132
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Intersection of subgroup of a free group with the lower central series
If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups
$\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\...
- 31
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79
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What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...
- 271
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267
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Definition of reducible lattice
I am reading Raghunathan's book on discrete subgroups of Lie groups.
In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
- 101
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97
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How large this subset is to say that it should equal the group?
Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set
$$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
- 2,118
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143
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About Jennings-Lazard-Zassenhaus series of groups
Let $G$ be a group and let $p$ be a fixed prime. For each positive integer $n$, the $n$-th term of the Jennings-Lazard-Zassenhaus series of the group $G$ is defined by the rule
\begin{eqnarray*}
D_{n}(...
- 77
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46
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Generalizing CIT-groups to odd case
A CIT-group is a group such that the centralizer of any involution is a 2-subgroup. The structure of these groups is known from the works of Suzuki and others.
Here is my question: has the odd case ...
- 307
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60
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Statement of a Theorem of H. Qu on number of subgroups in $p$-groups
The question is not much technical; I wanted to get to know the statement of a Theorem little clear. (I am not considering the proof of the Theorem).
Let $p$ always denote an odd prime, and $M_p$ is ...
- 1,169
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209
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Sylow subgroups of orthogonal group
According to Wikipedia (current revision) the cardinality of $O(n,q)$ depends on the properties of the field we're working over. These are the results:
We have the following formulas for the order ...
- 109
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126
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Combinatorics of merging sequences from multinomial coefficients
If you have $m$ sequences $a_{11},\dots,a_{1n_1}$ through $a_{m1},\dots,a_{mn_m}$ each sorted in ascending order (assume there are no duplicates) then there is an unique way to merge them.
How many ...
- 1,826
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186
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Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$
Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
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