Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,094 questions
12
votes
0
answers
373
views
Does Thompson's group $V$ have property AP?
Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
12
votes
0
answers
284
views
Star-shaped Folner sequence
Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that $...
- 4,432
12
votes
1
answer
744
views
Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 38.6k
12
votes
2
answers
1k
views
A variation of the Ryll-Nardzewski fixed point theorem
Is there a fixed-point theorem that implies the following result?
Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
- 45k
12
votes
3
answers
3k
views
Non-Lie Subgroups
A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof).
...
- 2,806
12
votes
5
answers
2k
views
Leech lattice decomposition
Hello,
I am investigating the Leech lattice. Lately I have discovered following. Some lattices decompose into distinct set of orthonormal frames. For example E8 lattice which contains 240 unitary ...
- 123
12
votes
4
answers
1k
views
How many non-isomorphic abelian subgroups of the permutation group $S_n$?
I am interested in how many (pairwise non-isomorphic) subgroups of the permutation group $S_n$ are abelian. ($n \in \mathbb{N}$ arbitrary and possibly big)
Are you aware of any references which treat ...
- 263
12
votes
0
answers
340
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
12
votes
2
answers
1k
views
Low-order symmetric group 2-generation: n=5,6,8
In a comment at the recent question What is the standard 2-generating set of the symmetric group good for?, it was remarked that the symmetric groups $S_n$ for $n\gt 2$, $n\neq 5,6,8$, can be ...
- 35.5k
12
votes
2
answers
890
views
Factorizable groups
Definition. A finite group $G$ is factorizable if for any positive integer numbers $a,b$ with $ab=|G|$ there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$.
Problem 1. ...
- 41.9k
12
votes
1
answer
553
views
Topological amenability vs amenability of an action
Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms.
Consider the following two definitions:
[$C^*$-algebras and finite dimensional ...
- 121
12
votes
3
answers
1k
views
Conjugacy in $GL(n,\mathbb Z)$
How can I determine whether $A_1,A_2\in GL(n,\mathbb Z)$ conjugate in $GL(n,\mathbb Z)$ and if they are, how can I find a $P\in GL(n,\mathbb Z)$ for which $A_2 = P^{-1}.A_1.P$ ?
In $GL(n,\mathbb Q)$ ...
- 347
11
votes
2
answers
574
views
Identifying a group without 2-torsion
Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there ...
- 18.7k
11
votes
1
answer
1k
views
When is an HNN-extension finitely presented?
Let $G=\langle H, t; K^t=K^{\prime}\rangle$ be an HNN-extension of $H$, with $t$ inducing the isomorphism $\phi: K\rightarrow K^{\prime}$. I was wondering if the following question can be answered, ...
- 2,821
11
votes
3
answers
594
views
Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with less than $n/2$ elements?
Say that $S\subset \mathbb Z_n$ is stabbed by $X\subset \mathbb Z_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$.
Is there for every $|S|=4$ an $|X|<n/2$ that stabs it?
My motivation ...
- 19k
11
votes
1
answer
868
views
Dessins d'enfants and absolute Galois group
I would like to know what is the recent progress about the group homomorphism
$$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$
$\mathrm{Gal}(\overline{\mathbf{...
- 1,613
11
votes
4
answers
2k
views
Iterated semi-direct products
Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where $...
- 1,661
11
votes
1
answer
550
views
Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the ...
- 9,501
11
votes
3
answers
1k
views
Congruence subgroups as abstract groups
This is probably well know, and maybe even trivial, but not to me. Consider for concreteness the subgroup
$$
\pm\Gamma_0(3)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}:\;a,b,c,d\in\...
- 13k
11
votes
2
answers
779
views
Characters of permutation groups
Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the ...
- 4,679
11
votes
2
answers
836
views
Cogroups in the category of groups are free
The free group $F(S)$ on a set $S$ is a cogroup in the category of groups since $\hom(F(S),G) \cong G^S$ carries a natural group structure for every group $G$. I have read that these are the only ...
- 63.1k
11
votes
1
answer
1k
views
Functorality of universal central extension
I'm looking a reference (or quick proof) for the following fact, which doesn't appear in the standard sources I've consulted (for instance, Milnor's book on algebraic k-theory).
Let $\phi : G \...
- 378
11
votes
0
answers
357
views
Recognizing reflection subgroups of Coxeter groups
Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
- 2,753
11
votes
4
answers
2k
views
Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
- 52.9k
11
votes
1
answer
739
views
Three involutions on the set of 6-box Young diagrams
The set of $n$-box Young diagrams classifies both conjugacy classes in $S_n$ and equivalence classes of irreducible representations of $S_n$. There is an outer automorphism of $S_6$, of order 2. ...
- 22.3k
11
votes
0
answers
541
views
Cyclic and prime factorizations of finite groups
A tuple $(A_1,\dots,A_n)$ of subsets of a finite group $G$ is called a factorization of $G$ if $G=A_1\cdots A_n$ and $|A_1|\cdots|A_n|=G$.
In Cryptology factorizations of groups are known as ...
- 41.9k
11
votes
1
answer
248
views
How many steps are required for double transitivity?
Let $A$ be a set of generators of $S_n$, or of a doubly transitive
subgroup of $S_n$. Assume $e\in A$, $A=A^{-1}$. What is the least $k$
such that $A^k$ is doubly transitive as a set? That is, what is ...
- 20.2k
11
votes
2
answers
932
views
A group action of the Heisenberg group with special symmetries
Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...
11
votes
1
answer
598
views
If $(F_n)_n$ is a Følner sequence satisfying Tempelman's condition, is $(F_n^{-1}F_n)_n$ also Følner?
Let $G$ be a countable group. A Følner sequence is a sequence of finite subsets $(F_n)_n$ such that
$$\lim_{n\to\infty} \frac{|KF_n \mathbin\triangle F_n|}{|F_n|} = 0$$
for each fixed finite subset $K ...
- 213
11
votes
0
answers
305
views
Detecting invertible elements in group rings by their images for finite quotients of the group
Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element ...
11
votes
2
answers
935
views
Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?
An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
- 2,147
11
votes
1
answer
357
views
Alternating subgroups of $\mathrm{SU}_n $
$\DeclareMathOperator\PSU{PSU}$Let $ \PSU_n $ be the projective unitary group. Let $ A_m $ be the alternating group on $ m $ letters.
$ A_5 $ is a maximal closed subgroup of $ PSU_2 \cong SO_3(\mathbb{...
- 4,081
11
votes
2
answers
495
views
A question about conjugacy in Higman's group
In his paper A finitely generated infinite simple group (J. London Math. Soc., 1951), Higman introduced the following finitely presented group:
$$
H = \langle x,y,z,w \mid [x,y]=y, \, [y,z]=z, \, [z,...
- 35.9k
11
votes
1
answer
302
views
Infinite vertex-transitive graph where every automorphism has a fixed vertex
This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof.
Let $G = (V,E)$ be a graph with $V$ infinite. ...
- 24.2k
11
votes
3
answers
2k
views
A categorical method to, say, determine the cardinality of a group
I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category ...
- 6,353
11
votes
2
answers
475
views
A sequence of subsets of an infinite group
Is there an infinite group $G$ such that there is not any sequence $(A_n)$ of its subsets such that always
$$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$
?
link
- 261
11
votes
2
answers
1k
views
Extension of the Hall-Witt identity
Although this question is mostly out of curiosity (as of now), I hope it is nevertheless suitable for MO.
This very recent (and still open) question about the Hall-Witt identity led me to wonder:
...
- 28.6k
11
votes
3
answers
1k
views
Centralizers in amalgamated free products
An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is ...
- 165
11
votes
9
answers
1k
views
Proving the impossibility of an embedding of categories
A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...
- 5,759
11
votes
1
answer
688
views
torsion group of the multiplicative group of a field
Let $F$ be any field of zero characteristic, $F^{\ast}$ its multiplicative group and $T$ is the torsion group.
Is it true that $T$ is a direct summand for $F^{\ast}$?
- 113
11
votes
1
answer
678
views
For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?
I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
- 25.8k
11
votes
0
answers
610
views
Group presentations with few relations and undecidable word problem
This question came up in an algebra class I'm teaching. It's not my field and I couldn't find an answer easily, so I thought I would ask it here.
Is the fewest number of relations in a presentation ...
- 1,601
10
votes
2
answers
2k
views
When is a Baumslag-Solitar group linear?
The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation
$BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
- 411
10
votes
1
answer
949
views
Generalizing the Notion of Nilpotent/Abelian/Cyclic Numbers
Let us call a number $n\in\mathbb{N}$ nilpotent if
$$n=p_1^{e_1}\cdots p_m^{e_m}$$
with $p_i^k\not\equiv 1\mod p_j$ for $i,j\in\{1,\ldots,m\}$ and $1\leqslant k\leqslant e_i$.
A cute theorem says the ...
- 777
10
votes
2
answers
941
views
Induced representations of topological groups
Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from https://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups)
If $G$ is a group ...
- 18.7k
10
votes
1
answer
821
views
How can you order a free group?
A left order on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A right order is defined symmetrically, and a biorder is an order that is ...
- 6,652
10
votes
0
answers
194
views
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 $n$ ...
- 7,545
10
votes
2
answers
1k
views
algorithm to compute the integral orthogonal group
Suppose I have an indefinite quadratic form over the integers, and I want to compute its orthogonal group. Is there an algorithm, or at least a heuristic? If yes, is there any implementation anywhere?
- 96.4k
10
votes
4
answers
1k
views
Twist of a group Hopf-algebra
Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
- 13.3k
10
votes
2
answers
538
views
Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups
Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
- 31.2k