Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,042
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abelian p- subgroups of E_6(q)
Is there any result about maximal abelian p-subgroups of the exceptional group E_6(q), where q=p^a is prime power?
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Complexity of decision problem to decide if permutation group is $k$-transitive
Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a ...
- 798
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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 $G$ ...
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How generic are Cayley graphs of non-Abelian groups with logarithmic girth?
Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...
- 1,863
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Zero divisors of the form $1+x+y$ in the rational group algebra
Is there a finite non-ablelian group $G$ generated by $x$ and $y$ such that $1+x+y$ is a zero divisor in the rational group algbera $\mathbb{Q}[G]$ and also $x^2$, $y^2$ and $(x^{-1} y)^2$ are all ...
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What are all the transitive extensions of cyclic groups?
"Let $G$ be a transitive group of permutations on a given set of letters. Let a new fixed letter be adjoined to every permutation of $G$. Then a transitive group $H$ of permutations on the ...
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If the union of finitely many conjugacy classes is syndetic, are there finitely many conjugacy classes?
(Cross-post from math.stackexchange.)
Let $G$ be a finitely-generated group. Write $A^G = \{g^{-1} a g \;|\; a \in A, g \in G\}$, and $A \Subset G \iff A \subset G \wedge |A| < \infty$. Is the ...
- 6,327
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Strongly graded algebras with no zero divisors
Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that
$$
m: A_i \otimes_{A_0} A_j \to A_{i+j}
$$
is an ...
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Does a cocompact subgroup of a topological group contain a cocompact normal subgroup?
Motivation: It is obvious that for a finite index subgroup $H$ of a group $G$, there exists a normal subgroup $K$ of $G$, $K\subset H$, with $|G/K|<\infty$.
Our question: Let $G$ be a topological ...
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Is $x + y \ne y+nx$ for $x \ne 0$ and $n \ge 2$ (in an ordered group)?
Let $(A, +, \preceq)$ be an ordered group, namely $(A, +)$ is a group and $\preceq$ is a total order on $A$ such that $x + z \prec y + z$ and $z + x \prec z + y$ for all $x,y,z \in A$ with $x \prec y$....
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Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,001
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centralizer of a n-cyclic permutation matrix over F_2 in GL(n,2)
This is a continuation of this question, where I talked about the case $n=2^k$. Let $C$ be the $n\times n$-permutation matrix over $\mathbb{F}_2$ of the $n$-cycle. We needed to know the explicit ...
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Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
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Groups arising as direct limits of a stationary system of primitive matrices over the integers
I am interested in the kinds of groups of the form $\displaystyle\lim_{\longrightarrow}(\mathbf{Z}^k,M)$ where $M$ is a primitive (some power of $M$ has strictly positive components) $k\times k$ ...
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Factor group of direct product by restricted direct product
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 $|F_{i}|&...
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The principal congruence subgroup of the symplectic group over the integers
Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
user341790
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Non-residually finite groups
Does anyone know groups which admit presentations with two more generators than relators and are not residually finite? If so, do we know anything about the finite residual of such groups?
Any ...
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Groups in which lower central series and upper central series coincide
Let $G$ a finite two-generated $p$-group in which lower and upper central series coincide. Clearly we obtain that the upper central series become strongly central, we have also that at least half of ...
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Type $C_n$ Weyl group contains in the centralizer of the longest word $w_0$ in $S_{2n}$
Are there some references about the proof of the following fact?
Type $C_n$ Weyl group lies in the centralizer of the longest word $w_0$ in $S_{2n}$.
Thank you very much.
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Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
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The sporadic numbers
Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the ...
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Group where Out(G) acts differently on conjugacy classes and irreps? [duplicate]
$\def\Conj{\mathrm{Conj}}\def\Irrep{\mathrm{Irrep}}\def\Out{\mathrm{Out}}$Let $G$ be a finite group, let $\Conj(G)$ be the set of conjugacy classes of $G$, let $\Irrep(G)$ be the set of isomorphism ...
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A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
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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(\...
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Are there overwhelmingly more finite posets than finite groups? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
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Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?
Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and $...
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simple groups all sylow subgroup is nonabelian
Thanks for any help or comments
How can I find the list of all non abelian simple groups (particularly simple lie type) such that all $p$-Sylow subgroups are non abelian for odd prime $p$?
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Infinite quotient of Hurwitz Group
I am currently working through all the groups with two generators, and I am up to the group with presentation $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9 \rangle$. I have found all the finite ...
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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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Trans-universality for finitely generated groups
QUESTION: does there exist a group U such that three conditions hold:
(a) every finitely generated group is isomorphic to a subgroup of U;
(b) for every group G that is not finitely generated there ...
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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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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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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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A question concerning some group action
Let $G$ be a finite group. Consider the set
$$X = \bigcup_{H \le G} G/H$$
which is a disjoint union of left cosets of subgroups $H$ of $G$.
Then $G$ acts on $X$ by left multiplication, and the number $...
user6671
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Examples of non-proper profinite HNN extensions
We define a profinite HNN extension as the profinite completion of the abstract HNN extension. In the abstract case, the homomorphim of the base group to the HNN extension is always a monomorphism. ...
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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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the number of minimal generating subsets of a group
Clearly every finite group has a minimal generating subset.
Is there any formula for the number of minimal generating subsets of a finite group?
Is it known which groups have a unique minimal ...
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The largest abelian subgroups of a Lie group
Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...
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Subgroups of powers of the alternating group on 5 elements
Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
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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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Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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