Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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Show me that I have not simplified the proof of the Adian-Rabin theorem
I am not a mathematics researcher but I am concerned that this question, posed with slightly different wording on math.stackexchange, may be too esoteric for that forum since it concerns the details ...
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short exact sequence of groups [closed]
We have a short exact sequence as
$$0 \rightarrow \mathbb{Z}_2\rightarrow G \rightarrow \mathbb{Z}_2\rightarrow 0,$$
can we conclude that the group $G$ is isomorphic to $\mathbb{Z}_2 + \mathbb{Z}_2$ ...
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Local triviality of torsors for relative reductive groups
Let $X \to S$ be a relative (smooth proper) curve, and $G \to X$ a reductive group scheme. The following two results are well-known:
(Drinfeld-Simpson) For arbitrary $S$, if $G$ is defined over $S$, ...
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The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
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What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
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Which algebraic groups are generated by (lifts of) reflections?
$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem
states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, ...
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Does every transitive permutation group contain a permutation whose cycle lengths have a common divisor?
Let $H$ be a transitive subgroup of $\mathfrak{S}_n$, $n \geq 2$.
Using Jordan's lemma ($H$ is not a union of conjugate proper subgroups), we see that $H$ contains a permutation without fixed points.
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Example of three dimensional atoroidal Poincaré duality group with some pathology
I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...
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What does this notation mean? [closed]
For context, $\phi_{f} $ and $\psi_{u} $ are two different automorphisms of the same group. I would like to know what the following notation, $\phi_{f}^{\psi_{u} } $ , is referring to? What does it ...
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Can automorphism equivalence in a free group be detected in a nilpotent quotient?
If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...
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Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
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Generators for the first cohomology of free groups
Let $F = \langle x_1, \dots, x_n \rangle$ be the free group on $n$ generators and $R = \mathbb Z$. The Fox derivatives $\frac{\partial}{\partial x_i} \colon F \to R[F]$ are the unique derivations ...
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Generalisation of abelianisation using representation theory?
This question didn't receive an answer on MathSE, so I'm asking it here.
Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic zero. Every $1$-dimensional ...
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An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
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What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
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Can a non-free Whitehead group embed as a discrete subgroup of a normed space?
Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
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Commutator of a group element on a vector space
I am reading a paper in which the author has a group $G$ admitting a representation $\pi$ on a vector space $V$. Let $g \in G$ be a group element. The author refers to a so-called "commutator of $...
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When are these irreducible complex representations for the Type D Weyl group self-dual?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
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Growth of powers of symmetric subsets in a finite group
(This question was originally asked on Math.SE, where it was answered in the abelian case)
Let $G$ be a finite group, and let $A$ be a symmetric subset of $G$ containing the identity (i.e., $A^{-1}=A$ ...
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Equivalence of dihedral and symmetric group actions on a specialized real algebra
Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...
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Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
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Large subgroups of Knuth's non-associative "group" on ${\cal P}(\mathbb{N})$
Donald Knuth introduced a fast, bit-wise approximation to integer addition by $$(a,b) \mapsto a \, ^{\land} \, b \, ^{\land} \, ((a \text{ & } b) \ll 1)$$
where $a,b$ are given in binary and $\,^{\...
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Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$ [closed]
Prove that every group $G$ with $p^n$ ($n\ge4$) elements and center with $p$ elements has an abelian subgroup of order $p^3$
I'm new in this forum so I hope I haven't made any mistake.
I have to ...
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Finitely generated G, such that x^3 = 1 for all x, is finite? [closed]
x^3 = e for any element x in finitely-generated group G. How to prove that G is finite?
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Divergence functions in hyperbolic groups
Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...
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Almost free group without the Specker group as a subgroup
An Abelian group is almost free whenever every countable subgroup is free Abelian. Famously, the Specker group $\mathbf Z^{\mathbf N}$ is almost free. What are examples of almost free groups that are ...
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Distances on spheres in Cayley graphs of non-amenable groups
Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...
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If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?
Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
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Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
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Finite 2-groups with $(ab)^{2}=(ba)^{2}$
There exist nonabelian finite 2-groups $G$ with the property $(A2)$ : for every $a,b\in G$, $(ab)^{2}=(ba)^{2}$. An example of a such group is given by the quaternion group $Q_{8}$ of order 8. Is ...
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Where can I find a table of the exponents of the sporadic groups?
Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties.
In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ...
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Finitely generated torsion-free pro-$p$ subgroup of ${\rm GL}_{n}(\mathbb{F}_{p}[[T]])$ is solvable?
Let $\mathbb{F}_{p}$ be a finite field of order $p$, and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. My question is the following:
Let $G$ be a closed pro-$p$ ...
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Normalizer of the group of segment $C^\infty$ diffeomorphisms in the group of segment homeomorphisms
What is the normalizer of the group of $C^\infty$ diffeomorphisms on $[0, 1]$, with group law given by composition, in the group of all homeomorphisms of $[0, 1]$?
If the answer is known, is there ...
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Conjugacy of Cartan subgroups in $\mathrm{GL}(n)$
$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary.
Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. ...
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Is this set, defined in terms of an irreducible representation, closed under inverses?
$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
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Normal subgroups of pure braid groups stable under strand bifurcation
$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...
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Finite groups with only one $p$-block
If $G$ is a finite group with a prime $p \big| |G|$, and $G$ has exactly one $p$-block, namely the principal block, can anything be said about the structure of $G$? I am aware that when $G$ has ...
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Just-infinite quotients of pro-$p$ groups that are linear over a complete Noetherian local ring
This question is a sequel to Quotients of pro-p
groups linear over a complete Noetherian local ring. Recall that an infinite pro-$p$ group is called just-infinite if it has no proper, infinite ...
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Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
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Number of conjugacy classes of a semi-direct product of two finite groups
Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\...
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A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups
Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...
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Is $SO(N)$ a sphere? [closed]
Is $SO(N)$ as a metric space with a Killing form metric, isometric to a sphere $S^M$ (for any $M$) with standard metric on the sphere?
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G-modules vs. $\Delta(NG)$-modules
Let X be a simplicial set. Its category of simplices, denoted by $\Delta(X)$, is the category whose objects are the pairs $(x,[n])$, with $x\in X_n$, and morphisms $\bar{c}:(y,[m])\to (x,[n])$, where $...
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Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
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Recognizing free groups
While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
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Research directions related to the Hilbert-Smith conjecture
The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
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Do balls in expander graphs have small expansion?
Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...
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Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
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Groups acting non-properly cocompactly on hyperbolic spaces
A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...
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Groups (not necessarily finite) with a given number of maximal subgroups
It is somewhat easy to see that a group $G$ with exactly one maximal subgroup $M$ must be cyclic: any element in $G\setminus M$ generates $G$.
EDIT: @YCor pointed out in the comments that this ...
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