Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Are almost all permutation configurations from $S_n$ covered by small subsets subgroups of $S_n$?
Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$.
Do we have ...
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Lattices are not solvable in non-compact semisimple Lie groups
I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
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Centralizers of Cartan subgroup III
Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Let $n$ be a positive integer. The multiplicative group $(\mathcal O/n\mathcal O)^\times$ acts on the module $\mathcal O/n\mathcal O\...
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Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups
First of of all I'm trying to find a general interpretation to the following facts below.
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
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Product of two group morphisms not a group morphism
In Mac Lane's Categories for the Working Mathematician, page 110, second edition, he states that, in the category of groups $Grp$, $F_n$ being groups and $x_n, y_n$ being two cones on these groups (...
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$\omega$-nilpotent cover of a recurrent surface
Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville.
$\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
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Find representation set of orbits when group acts on a set
Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....
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Unit-product sets in finite decomposable sets in groups
A non-empty subset $D$ of a group is called decomposable if each element $x\in D$ can be written as the product $x=yz$ for some $y,z\in D$.
Problem. Let $D$ be a finite decomposable subset of a ...
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A cross product on $C^*_{red} G$
For every group $G$, the reduced group $C^*$-algebra $C^*_{red}G$ is equipped with the inner product $\langle a,b\rangle=tr(ab^*)$ where "$tr$" is the standard trace on group $C^*$-algebras.
For ...
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On equality of two quotients of a congruence subgroup
Related question: Non-torsion part of the abelianisation of congruence subgroups
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
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Pure (ordered) subgroups
Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
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An alternative proof of a subgroup lattice characterization of the infinite cyclic group
In Schmidt's book Subgroup lattices of groups, Theorem 1.2.5 states that a group $G$ is cyclic if and only if its subgroup lattice $L(G)$ is distributive and satisfies the maximal condition. Its proof ...
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Example of a spherical homogeneous space $G/H$ with a pairs of colors and with the center of $G$ not contained in $H$?
Let $G$ be a simply connected simple algebraic group over $\mathbb C$,
$B\subset G$ a Borel subgroup, and $T\subset B$ a maximal torus.
Let $\mathcal{S}=\mathcal{S}(G,T,B)$ denote the set of simple ...
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A question on UCS p-groups(2)
This is a follow up to this question
Let $G_{1}$ and $G_{2}$ be two finite UCS p-groups with the following conditions:
1- $\vert G_{1}\vert=\vert G_{2}\vert=p^{2n}$;
2- $\Phi(G_{1})\cong\Phi(G_{2})\...
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Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
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Computing the class-preserving automorphism group of finite $p$-groups
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called a class-preserving if for each $x\in G$, there exists an element $g_x\in G$ such that $\alpha(...
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Non-zero homomorphism from a module to its ground ring
Let $c_1,\dots,c_k$ be some non-zero complex numbers and $M$ be the abelian subgroup generated by $c_1,\dots,c_k$ (i.e. all $\mathbb{Z}$-linear combinations of $c_1\dots,c_k$). Suppose further that $\...
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Is the set of fixed points of hyperbolic elements from $\mathrm{PSL}_{2}(A)$ a group?
Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal ...
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On $n$th class-preserving automorphism of finite $p$-group
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. An automorphism $\alpha$ of $G$ is called an $n$th class-preserving if for each $x\in G$, there exists an element $g_x\in \gamma_n(G)$ ...
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Defect of subnormality in unit groups of modular group algebras
Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...
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Relations between $\Omega$-groups, locally indicable groups, and right-orderable groups
We know that the class of right-orderable groups $\mathit{RO}$, is contained in the class of $\Omega$-groups (read it from "A note on group rings of certain torsion-free groups" by Burns-Hale).
A ...
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Irreducible characters of a semi-direct product with a p-group
Suppose G is a semi-direct product of P with H where P is a (non-abelian) p-group and G is solvable. I wonder what can be said about the irreducible characters of G given information about the ...
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Unitary element of the group algebra
Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
- 2,118
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What is an upper limit of relative size of conjugacy class of the transitive finite group?
What is
$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$
$G$ transitive permutation group?
And what are the ...
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Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra
Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...
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Walk in the graph induced by a group action
Suppose that graph $G$ is induced by a group $⟨α_1,...,α_r⟩$ acting on a large finite set $X$ for small $r$. To be precise, we have the vertex set $V(G):=X$,
and $x_1x_2\in E(G)$ whenever for some $\...
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Some questions on linear algebraic groups and their eigenvalues
Let $G$ be a connected linear algebraic group (a torus, for example) of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups.
Suppose ...
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Tensor product decomposition of commuting representations
If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
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Functoriality in the group $G$ of the domain of the Baum-Connes map
Lück claims in his preliminary book, that the left hand side of the Baum-Connes map is functorial in the group $G$. For the right hand side $K(A \rtimes G)$ this is clear for the full crossed product, ...
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about a strange property of p-groups of maximal class
I am trying to look for a finite $p$-group of maximal class of order at least $p^{2p+1}$ exponent at least $p^3$ which possibly has the following property :
If s is an element in $G-G_1$ ($G_1$ is ...
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Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
- 1,283
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Character degrees of a finite group?
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
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The finite extensions of $SL_2(q)$ [closed]
Let $G$ be a finite group such that $Z(G)\leq SL_2(q) \leq G $ and $G/Z(G) \cong PGL_2(q)$· Is there any information about the structure of $G$?
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Counting conjugacy classes with a subgroup of prime index
I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a ...
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In 3D point groups, does $[\Gamma_{e}\otimes\Gamma_e] = \Gamma_{Rot_z} \forall$ degenerate $\Gamma_e$ hold in general?
In the following I am referring to groups exclusively describing 3D point symmetries. I use the Schönflies notation for groups and their elements and the Mulliken symbols to describe their irreducible ...
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Isomorphism in homology
I asked this question on Mathematics SE three days ago, but didn't get the answer.
$\require{AMScd}$Let $G, H, K$ be groups and suppose that we have a diagram
$$\begin{CD}
G @>f_1>> H\\
@...
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A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
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A functor on the category of commutative rings, algebras or Banach algebras
Edit: According to the comments of abx and Yemon Choi I revise the question as follows:
Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...
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Conjugacy classes of non-normal subgroups of a finite $p$-group
Let $G$ be a finite $p$-group of derived length $d$ and nilpotency class $c$. Suppose that $G$ is not a Dedekind group (i.e., possesses at least one non-normal subgroup). Suppose that $G^{(d-1)}$ has ...
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partially commutative like monoids [duplicate]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\}...
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partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
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Is there a perfect group which is a finite extension of the discrete Heisenberg group $H_3(\Bbb Z)$? [closed]
We assume that $G$ is a finite extension of the discrete Heisenberg group $H_3(\Bbb Z)$, that is,
$$
1 \rightarrow H_3(\Bbb Z) \rightarrow G \rightarrow F \rightarrow 1
$$
where $F$ is a finite group.
...
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Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
- 1,959
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Alternating Hurwitz quotients multiplicity
How many times is each alternating group a hurwitz quotient? In other words, how many non-isomorphic ways can an alternating group be generated by an element of order 2 and an element of order 3 whose ...
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Is there a decomposition exists for $e^{c(K_++K_-)^2}$
In the usual $SU(1,1)$ group:
$$[K_+,K_-]=-2K_z,\quad [K_z,K_\pm]=\pm K_\pm.$$
Is there a decomposition exist for $e^{c(K_++K_-)^2}$?
Of course there won't exist a decomposition to $e^{K_+},e^{K_-},...
user120129
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Divisible group as the automorphism group of a algebraic variety?
Let $X$ a affine algebraic variety. Then $X$ can have infinite automorphism group, for example $X=\mathbb{A}^1$. Let $G$ be a divisible abelian group.
My question is about some condition in $G$ such ...
- 527
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Generalisation of the Liouville function as irreducible representations for the semigroup ($\mathbb{N},\cdot)$?
This is a duplicate of a question I have asked at here at math stack exchange, but I thought it could be also here of interest.
When looking at the [Liouville function] (https://en.wikipedia.org/wiki/...
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102
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Efficient characterisation of the generalized symmetric groups
This is a follow-up to normal form for some finite groups, extending the small groups library.
Not being familiar with groups, I wonder whether it is possible to check efficiently whether a group (...
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On groups with finite pro-$p$ completion for all primes $p$
Say that a group has Property X if its pro-$p$-completion is finite for every prime $p$. For instance, every perfect group has Property X.
Is there a finitely generated, residually finite group $G$ ...
- 5,450
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139
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Direct product of hopfian groups
This is a question about Hirshon's paper "The center and the commutator subgroup in hopfian groups"(https://projecteuclid.org/euclid.afm/1485894451). Theorem 12 stating that if $B$ is a perfect group ...
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