Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
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For a group with one end does the property of connected spheres follow?
One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group $G$ with generators $g_1, \ldots g_n$....
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Lacunary hyperbolic groups and weak amenability
In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
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Centralizers of group actions
Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...
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is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
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Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog
I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...
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Where can I find analogues of combinatorial central limit theorems for other groups
The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum:
$$\displaystyle f(\...
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A uniform bound for a "true" non-congruence subgroup
Before stating my question, let me recall the Congruence Subgroup Property/Problem: Given simply connected absolutely and almost simple algebraic group $G$ with fixed realization as a matrix group one ...
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A subgroup of corank 1 in a free group contains a primitive element?
Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq \...
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Cyclic numbers of the form $2^n + 1$
A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
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Is there a strictly coassociative resolution of polynomial growth, for a finite group?
Let $G$ be a finite group and $k$ a field of characteristic $p$. It is well known, thanks to the work of Quillen, that the trivial $kG$-module $k$ has a projective resolution of polynomial growth. To ...
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Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
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Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$.
Explicit formulas with formal ...
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Dual Coxeter numbers, Langlands dual groups, black holes and twisted compactification of 6d (2,0) A D E theories on a circle
A 6-dimensional (2,0) superconformal quantum field theory comes in Lie algebra A, D, E types. These theories do not have classical Lagrangian and are purely quantum.These theories on a torus ...
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Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)
I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
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Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...
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Hopficity of Baumslag-Solitar groups
I am struggling to find the exact source (with proofs) of the following ''well-known'' statement:
the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...
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Decidable membership for subgroup generated by three elements in $F_2\times F_2$
Let $F_2$ be the non-abelian free group on two generators. Let $G\subset F_2\times F_2$ be a subgroup generated by three elements. Is there an algorithm deciding if a given element of $F_2\times F_2$ ...
user178109
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Algebraic context for Mednykh's formula?
Let $S$ be a closed orientable surface and let $G$ be a finite group, then Mednykh's formula says that
$$
\sum_{V}d(V)^{\chi(S)} = |G|^{\chi(S) - 1} |\text{Hom}(\pi_1 S, G)|
$$
where the sum is over ...
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Groups with trivial outer automorphism group and prescribed center?
Given an arbitrary abelian group $A$, can we find a group $G$ such that
$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and
$Z(G)\simeq A$?
Why is this interesting? Given a group $G$, we have ...
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Infinite-dimensional torsion-free $F_\infty$-group not containing $F$
Is there an example of a group $G$ that has the properties
the cohomological dimension of $G$ is infinite: $\operatorname{cd}(G) = \infty$,
$G$ is torsion-free,
$G$ is of type $F_\infty$,
$G$ does ...
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Which finite solvable groups have solvable automorphism groups?
Is it possible to give a reasonable description of those finite solvable groups $G$ such that $A = {\rm Aut}(G)$ is also solvable?
The central case to deal with is that in which $G$ is a $p$-group of ...
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Sufficient condition for complementation of abelian normal subgroup
Suppose that we are given a finite $p$-group $G$ and an abelian normal subgroup $A$ of $G$. The question I have is whether any sufficient conditions are known for $A$ to have a complement in $G$. From ...
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Uncountable group with no proper subgroup of maximal cardinal
The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper ...
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Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
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Colimit of continuous cohomology over subgroups
Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
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Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
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Permutation groups with diameter $O(n \log n)$
I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture:
Suppose that
1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...
- 5,847
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Hochschild-Serre spectral sequence via explicit filtration
Let
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$$
be a short exact sequence of groups and let $M$ be a $\mathbb{Z}[G]$-module. The Hochschild--Serre spectral ...
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Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?
First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...
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Sharpness of the $1/6$-constant in the Cancellation Theorem
I originally posted this question over at Stackexchange, before realising it is much better suited for Overflow:
Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $...
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Decomposition of linear groups into free products
I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
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A robust version of Schur's lemma?
Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
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$p$-groups and the arithmetic of $p$
I cannot think of an example of a property of a $p$-group or a pro-$p$ group that depends on the arithmetic of the prime number $p$. To clarify l do not mean the size of $p$. Clearly lots of stuff ...
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Using Property (T) to approximate invertible matrices
In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:
Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can ...
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Generating $S_n$ with a fundamental transposition and a big cycle
I apologize in advance if this is too amateur, this is not really my area, but I'm very curious.
We have a permutation $\pi \in S_n$ and we want to represent it as a product of $\sigma = (1\;2)$ and $...
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(Torsion in) homology of free nilpotent groups
It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
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When is the rank 2 free metabelian group of exponent $n$ center free?
Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
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"A remarkable Moufang loop"
The 1985 paper A simple construction of the Fischer-Griess monster group by Conway refers to an "in press" article, A remarkable Moufang loop, with an application to the Fischer group $Fi_{24}$, by ...
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pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
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Derived functor of Malcev completion ( = "rational K-theory")
By a theorem of Keune (which is more of definition rather then theorem per se) K-groups of ring $R$ are Puppe's derived functors of pronilpotent completion of $GL(R)$. They are derived in the most ...
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Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
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Is there an countable amenable dense subgroup of $U(\ell^2 \mathbb N)$?
Question: Does the unitary group $U(\ell^2 \mathbb N)$, equipped with the strong operator topology, contain a countable dense subgroup which is amenable as a discrete group?
I would be also ...
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When does a CAT(0) group contain a rank one isometry
Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $...
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Tarski Monster group with prime 5
Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about 5?
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Group with unsolvable conjugacy problem but solvable conjugacy length?
Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its ...
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Good bounds for the number of $n$-dimensional crystallographic groups ?
Let $s(n)$ denote the number of distinct crystallographic groups in $Isom(\mathbb{R}^n)$.
Apparently the best known upper bound so far is
$$
s(n)\le e^{e^{4n^2}},
$$
given by Peter Buser in $1985$. On ...
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'Almost-isomorphic' groups
What can be said about pairs of non-isomorphic groups which are epimorphic images of one another and which also embed into one another?
Can such pairs of groups be 'classified' in some sufficiently ...
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'Infinitesimal' elements of a topological group
Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...
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Reflection groups in O(n+1,n) arising `in nature'?
For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...
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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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