Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Chain of automorphism groups
Let $\mathrm{Aut}(\Gamma)$ be the automorphism group of a graph $\Gamma$. Also suppose that $\mathrm{Cay}(G,S)$ is the Cayley graph of a group $G$ with respect to the generating set $S$. Consider the ...
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How to determine if a set is a sumset
Let $G$ be a commutative group (assume whatever you want on $G$ if needed. I am mainly interested in $G = \mathbb{Z}/n\mathbb{Z}$).
Let $k$ be a fixed integer.
Let $(a_1, \dots, a_{k^2})$ be a list of ...
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Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? [closed]
I asked this question on MSE here
This question was inspired by: The influence of conjugacy class sizes on the
structure of finite groups.
My question is as follows: Is there a way to study the ...
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A question on irreducible affine Coxeter groups
I have a question about affine Coxeter groups when reading Humphreys's book:
Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be ...
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generator of a subring of integral group ring
Let $G$ be a cyclic group of order $n$ and $K\leq AutG$ be a subgroup of the automorphism group of $G$. We denote the orbits of the natural action of $K$ on $G$ by $O_1,\cdots, O_s$. Let $\underline{X}...
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Abelian groups and their subgroups
It is well known that every finite abelian group is a direct product of cyclic groups. So for every $n$ every finite abelian group of exponent $n$ is a direct product of cyclic groups of order at most ...
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Recognition of finite simple groups by number of Sylow p-subgroups
Let $G$ and $G'$ be two finite simple groups and $p$ be a prime divisor of $\vert G\vert$ and $\vert G'\vert$. Also suppose that every Sylow p-subgroup of $G$ and $G'$ is a prime order subgroup($C_{p}$...
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Generalisation of a loop concept
Suppose that $(M, \circ)$ is a set $M$ over which there is defined a binary operation $\circ$ so that we have:
1) For every $(a,b) \in M \times M$ we have $a \circ b \in M$
2) For every $a \in M$ ...
user114642
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Assumption in Fulton & Harris? (representations of $SL_2(q)$)
(Originally asked at https://math.stackexchange.com/questions/2399091/restricted-representations-and-irreducibility-in-sl-2q)
I'm working through the Fulton-Harris ad-hoc method of constructing the ...
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Subgroup of free profinite group is free profinite?
The question is already in the title.
It is known that any subgroup of a free group is free. My question is:
Is a closed subgroup of a free profinite group is again a free profinite group ?
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Existence of an orbit of exponential growth for group acting on the real line
Let G be a non-abelian finitely generated subgroup of increasing homeomorphisms of the real line having a fixed point free element $h$ ($hx>x$ for all $x$ in the line). Is there a real number $a$ ...
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Groups with $G^n \cong G$ for some integer $n$ [duplicate]
Which integers $n>2$ have the following property?
There is a group $G$ such that
$G^n \cong G$; and
for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.
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quasiprimitive non-solvable groups
I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
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finite index, self-normalizing subgroup of $F_2$ [closed]
Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$.
Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that $...
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Conjugate in the symmetric groups
Hi all,
Let $p=2^n+1$ be an arbitrary (Fermat) prime. Consider $\pi=(1,2,...,p)\in S_p$, the symmetric group of degree $p$. My question is:
Is there always a cycle, denoted by $\rho$, of length $p-1$...
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Suzuki group order
Let $^{2}B_{2}(q)$ be Suzuki simple group where $q=2^{2n+1}$. I want to know order of two Suzuki simple groups can divide each other? In other words, suppose that $|^{2}B_{2}(q_{1})|\mid |^{2}B_{2}(q_{...
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Non-trivial action of $SL_n(\mathbb{Z} )$ on a simplicial tree
A group $G$ has Serre's property $FA$ if any isometric action of $G$ on a simplicial tree has a global fixed point. Let $n\geq 3$. It is well-known that $SL_n(\mathbb{Z} )$ has property $FA$. Now my ...
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Changing the Type of a Module in MAGMA
I am currently working with irreducible $k[G]$-modules in MAGMA for finite fields $k$ and finite groups $G$. To construct these modules, I am using the commands IrreducibleModules(G,k) This results in ...
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Homotopy Equivalence of Posets for the Weyl Group
How do I go about proving Quillen's Homotopy Equivalence of Chevalley (subgroups of) automorphism groups of finite classical Lie algebras. Given $\sigma$ , such that Chevalley’s construction $GU$$n$($...
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No non-trivial homomorphism to a group
Here is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question;
Let $G$ be a compact Hausdorff ...
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For the symmetric group on an infinite set, is there a generating set of strictly smaller cardinality? [closed]
Let $S_{\kappa}$ denote the symmetric group on some set of cardinality $\kappa$. Does there exist a generating set $X \subset S_{\kappa}$ such that $|X| < |S_{\kappa}|$ ($\stackrel{?}{=} 2^{\kappa}$...
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Intersections of conjugates of the icosahedral group in SO(3)
(Related question)
Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is ...
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Is there an Isomorphism between R and C under addition? [duplicate]
Possible Duplicate:
AC in group isomorphism between R and R^2
Somewhere, I recall being told that there is an isomorphism between $\mathbb{R}$ and $\mathbb{C}$ under addition. However, despite a ...
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group action and orbit space
suppose group G acts on group W,i.e.there is an injective hom from G to Aut(W).
different injections give different actions.if the orbit spaces of two G actions on W are the same,on what ocassions, do ...
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Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
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Commutator group and conjugacy classes
Let $G$ be a finite solvable group which is not nilpotent, and let $H=[G,G]$ be the commutator subgroup of $G$. Does the following hold for $G$ and $H$?
"There exists $g \in G \setminus H$ and $h ...
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Are all "almost projective" groups free?
Let us say that a group $H$ is almost projective if, given any group epimorphism $f\colon G\to H$, there is an embedding $i\colon H\to G$.
Does it follow that $H$ is free? If not, is there a ...
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Topological generators for the Sylow pro-$p$ subgroup of $\mathrm{SL}_2(\mathbf{Z}_p)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $G_2(\mathbf{Z}_p):=\begin{pmatrix}
1+p\mathbf{Z}_{p} & \mathbf{Z}_{p}\\
p\mathbf{Z}_{p} & 1+p \mathbf{Z}_{p}
\end{pmatrix}$. ...
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Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
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Lower bound of the largest irreducible character degree of alternating group $A_n$
$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
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Coboundary operators, 1-cocycles and computing cohomology
My question is about the compatibility and consistency between two definitions of cohomology in two books.
I asked this question about 10 days ago on MathSE
and I set a bounty on it, but I didn't ...
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Dimension of projective cover of trivial $kG$-module
Given a field $k$ with characteristic $p$, let $G$ be a transitive permutation group on $4p$ points. Let $P$ be a Sylow $p$-subgroup of $G$ and $Q\leq P$ is a $p$-subgroup of $P$ of index $p$. Now ...
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Is there a $G$-paradoxical $G$-invariant subset of the plane for $G$ a group of rigid motions?
The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $...
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Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group
I am reading a paper which makes the following claim:
let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$.
Let $X' = X \vee S^2$ be the wedge sum of $X$ with the ...
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Chains of numbers generated by 2 involutions
$\DeclareMathOperator\GF{GF}$Consider the finite field $\GF(p)$ for prime $p$.
Consider the pair of involutions $f(x) = 1-x$ , $g(x) = 1/x$, and the chain of numbers generated by these 2 involutions ...
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Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
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Definition of the Gauss symbol [closed]
Tahara refers to the "Gauss symbol" in the article, On the second cohomology groups of semidirect products, Math. Z. 129 (1972) 365--379. For a fixed $n$, let $S_{ij}$ be the expression
\begin{...
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Elusive groups and vertex-transitive graphs
This question is pertaining to finite connected vertex-transitive graphs.
I recently read "Transitive permutation groups without semiregular subgroup" by Cameron, Giudici, Jones, Kantor, Klin, ...
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Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]
Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
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Ordered group acting freely on partially ordered set
Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s_1 &...
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How many pairs of numbers between 0 and n-1 are equal to z mod n? [closed]
I want to know how to compute this function:
$f : \mathbb{Z}_m \rightarrow \mathbb{N}$
$f(z) = |\{ (x, y) \in \mathbb{Z}_m^2 \mid xy \equiv z \}|$
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Trivialize a cup-product 2-cocycle of $G$ in a larger group $J$
I like to ask a simple question: how to trivialize a cup-product 2-cocycle of $G$ into a 2-coboundary of $J$ in a larger group $J$.
Let us take a nontrivial 2-cocycle $\omega_3^G(g_a, g_b) \in H^2(G,\...
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Reference for 'Normal Subgroups of Fuchsian Groups'
I am looking for a reference on how to explicitly construct normal subgroups of a given Fuchsian group. I appreciate any help.
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Are all profinite groups conjugacy separable?
Let $G$ be an infinite profinite group.
Then $G$ is an inverse limit of an inverse limit system $\{G_i\}_{i=1}^{\infty}$ where $G_{i+1} \to G_i$ is surjective for all $i$.
I want to show that $G$ is ...
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The relationship between $p$-solvable Group and solvable group [closed]
Can anyone please tell me The relationship between $p$-solvable Group
and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.
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Perfect $Q[G]$-complex
Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex.
Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to $...
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Is the Frattini subgroup of a f.g virtually pro-p group open?
Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq_o G$. Must $G$ have only finitely many maximal open subgroups?
...
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Example of a polycyclic group which is not of polynomial growth? [closed]
The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?
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A question on direct limits of finite $p$-groups
Where can we find a well developed material on direct limits of finite $p$-groups?
For instance, is there a characterization of such groups, which have a finite rank (that is every subgroup can be ...
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Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$
Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
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