Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,095 questions
14
votes
2
answers
1k
views
The maximum order of finite subgroups in $GL(n,Q)$
For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders.
It appear in many articles a ...
- 2,829
14
votes
2
answers
400
views
A finite group that has no decomposition of given cardinality
Let $a,b$ be two positive integer numbers. A group $G$ is called $a{\times}b$-decomposable if there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$ where $AB=\{xy:x\in A,...
- 42k
14
votes
2
answers
985
views
Has the Jacobson/ Baer radical of a group been studied?
On groupprops, the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which ...
14
votes
1
answer
790
views
Order of elements
Consider natural numbers $m,n,k > 1$. There are finite groups $G$ containing elements $x,y$ such that $o(x) = m, o(y) = n$ and $o(xy) = k$. After embedding these groups in $S_\mathbb{N}$ we drive: ...
user30378
14
votes
2
answers
1k
views
Random walks on Coxeter groups
Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then
$$f_N(t)=\frac{1+2t+2t^2+t^3}{1-Mt-Mt^2+\frac{...
- 1,363
14
votes
2
answers
962
views
Groups which are only defined up to conjugation
I'm trying to understand what the right way is to think about "groups which are only well-defined up to conjugation." Since this is somewhat vague let me clarify it by pointing out the main examples ...
- 28.1k
14
votes
2
answers
593
views
What is known about the structure of finite groups admitting an automorphism where all elements have "norm" one?
Let $G$ be a finite group admitting an automorphism $\sigma$ of prime order $p$. Define the norm map $N:G\rightarrow G$ with respect to $\sigma$ by $N(g)= g\sigma(g)\sigma^2(g)\dotsb\sigma^{p-1}(g)$.
...
- 1,949
14
votes
1
answer
861
views
When is non amenablity witnessed by a single non measurable set?
Suppose $G$ is a finitely generated discrete group and that there is a subset $E$ of $G$ such that if
$\mu$ is a finitely additive probability measure on $G$, then there is a
$g$ in $G$ such that $\mu(...
- 3,547
14
votes
4
answers
631
views
Normal subgroups of an extension of the Higman group
Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ acts on $H_4$ in the obvious way (permuting the four standard generators cyclicly). The group ...
- 20.2k
14
votes
3
answers
2k
views
Minimal generating sets of groups
I am not exactly a group theorist, so this may be well-known.
Let $G$ be a finitely generated group such that the cardinality of minimal generating sets of $G$ is bounded above.
Does it follow that $...
- 1,072
14
votes
2
answers
2k
views
Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free
My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer:
Open question: Does there exist a finitely generated Zariski-dense torsion-...
- 2,374
14
votes
2
answers
1k
views
Obstructions for a group to be the multiplicative group of a field [duplicate]
It is well known that every finite multiplicative subgroup of a field is cyclic.
I somehow got interested in a possible reverse implication:
Assume we have an abelian group $G$ whose every finite ...
- 6,741
14
votes
2
answers
789
views
Restriction of a branched cover to its branch locus
Assume that we have a smooth, compact, complex surface $X$, and a smooth and irreducible divisor $B \subset X$. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G,...
- 66.3k
13
votes
2
answers
675
views
Laws characterizing the trivial group
What are laws characterizing the trivial group? I mean all group words $w$ such that if the identity $w=1$ holds in a group $G$, then $G=1$.
For example, it can be easily verified that if a group ...
- 2,233
13
votes
1
answer
2k
views
Some questions about the Malcev completion
Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
- 4,484
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
13
votes
1
answer
398
views
Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?
Motivation
The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...
- 2,549
13
votes
1
answer
370
views
Factorizing groups into a product of solvable subgroups
Does every finite group $G$ have a factorization $G=H_1\cdots H_k$ where the $H_i$ for $1\le i\le k$ are solvable subgroups of $G$ and $|G|=|H_1|\cdots |H_k|$ (equivalently, every element of $G$ is ...
- 787
13
votes
1
answer
458
views
Locally finite groups containing all finite groups
Say that a group is rich if it contains isomorphic copies of all finite groups.
It is easy to produce rich groups, and also rich locally finite groups, for instance the restricted direct product $A=\...
- 63.9k
13
votes
1
answer
466
views
Can matrices over $F[G]$ have infinite order when $G$ is periodic?
Let $G$ be a group, $F$ a field and $d \geq 1$. When does the ring $M_d(F[G])$ of $d$-by-$d$ matrices over the group ring $F[G]$ admit an
element of infinite multiplicative order? When does it admit ...
- 6,652
13
votes
2
answers
498
views
Decidability of word problem for group admitting certain action
Let $G$ be a group acting highly transitively (and faithfully) on a set $S$. Suppose that $G$ is finitely presented, and that every stabilizer in $G$ of a finite subset of $S$ is finitely generated. I ...
- 3,740
13
votes
1
answer
383
views
Does this group construction preserve finite presentability?
Suppose $G$ is a group. Consider the set $G^G$ of all functions $G \to G$, which forms a group under elementwise multiplication. Now, for all $g \in G$ let’s define $c_g \in G^G$ as the constant ...
- 2,618
13
votes
0
answers
387
views
On sentences true in all finite groups (revisited)
Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$
I am interested in the following questions.
(1) Is the sentence $(\forall\bar ...
- 893
13
votes
3
answers
946
views
Example of reflective subcategory of (Groups) whose reflector doesn't preserve finite products
A subcategory $D$ of a category $C$ is called reflective, if the embedding $D \hookrightarrow C$ has a left adjoint $L:C \to D$. The left adjoint $L$ is called the reflector. If the category $C$ is ...
- 1,484
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
13
votes
0
answers
1k
views
Cyclic Sylow $p$-subgroup in finite simple groups
I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt.
All finite simple groups have a cyclic
Sylow $p$-subgroup for some $p$
I found ...
- 2,829
13
votes
1
answer
459
views
A generalization of residual finiteness to topological groups
Consider the following generalization of residual finiteness to
topological groups.
A locally compact Hausdorff group $G$ is called residually compact if
for every compact $K \subseteq G$ there is a ...
13
votes
2
answers
454
views
Linear occurrences of finite simple groups
Let $S$ be a finite simple group. All representations below are over the complex numbers.
Let
$d_0(S)$ be the smallest dimension of a faithful representation of $S$,
$d_1(S)$ be the smallest ...
- 63.9k
13
votes
1
answer
1k
views
Iterated Automorphism Groups
Notation: For each group $G$ define:
$Aut^{(0)}(G):=G$
$Aut^{(1)}(G):=Aut(G)$
$\forall n\geq 1~~~Aut^{(n+1)}(G):=Aut(Aut^{(n)}(G))$
Question: Consider $I\subseteq \omega$. Is there a group $G$ ...
user43940
13
votes
1
answer
455
views
Variety of nilpotent Lie algebras or $p$-groups
Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...
- 63.9k
13
votes
1
answer
791
views
How nearly abelian are nilpotent groups?
It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2
Can this sentiment be made precise
in the sense of the
Turán and Erdős definition of "the probability that two elements of ...
- 151k
13
votes
2
answers
1k
views
Action of SL(2,Z) on upper triangular primitive integer matrices of determinant N, from the right. Is it transitive?
I am porting this question across from StackExchange, since it has received no answers and perhaps is sufficiently deep to fit here.
I am considering the set of upper triangular matrices
$$D_N=\left\...
- 133
13
votes
3
answers
882
views
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
13
votes
3
answers
933
views
Probability of commutation in a compact group
It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.
If instead $K$ is a compact group,...
- 52.3k
13
votes
4
answers
1k
views
Can we bound degrees of complex irreps in terms of the average conjugacy class size?
This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and ...
- 25.8k
13
votes
3
answers
900
views
"Big" groups $G$ with trivial $Out(G)$
We are looking for examples of groups $G$ such that $G$ is "big", but $Out(G)$ is trivial. By "big" we mean things like virtually free, or large, or Golod-Shafarevich. However, we would like our ...
- 5,450
13
votes
1
answer
1k
views
Difference between the completed group algebra and the profinite completion of a group ring
Let $G$ be a reasonably nice group, say residually finite if need be.
We may consider the group algebra $\mathbb{Z}[G]$.
Let $\widehat{\mathbb{Z}[G]} := \varprojlim_I\mathbb{Z}[G]/I$ be the ...
- 7,527
13
votes
2
answers
1k
views
What is the universal property of the Weyl group?
If $G$ is a group and $H\le G$ a subgroup, let $NH$ denote the normalizer of $H$ in $G$, and let $WH = NH/H$; following May's Concise Course, §3.4 this I call the Weyl group. I have also seen the ...
- 1,838
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
3
answers
2k
views
Generators for SL_2(R) for rings of integers R
Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?
...
- 270
12
votes
2
answers
1k
views
Cohomological dimension of a homomorphism
Let $G$ and $\Gamma$ be discrete groups, and let $\phi\colon\thinspace G\to \Gamma$ be a homomorphism.
Define its cohomological dimension $\operatorname{cd}\phi$ to be the least integer $d$ such that $...
- 35.9k
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
2
answers
583
views
Do there exist acyclic simple groups of arbitrarily large cardinality?
Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$.
In ...
- 64k
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
1
answer
720
views
Verify that a group is hyperbolic via computer algebra
I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if ...
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
474
views
"Conjugacy classes–irreducibles" bijection, but for permutation representations
The linear representation theory (over say $\mathbb{C}$ for concreteness) of a finite group $G$ is "the same as" its character theory. Characters are naturally functions on conjugacy classes of ...
- 24.2k
12
votes
0
answers
424
views
Reference for class of involutions containing longest element of finite Coxeter group?
Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
- 52.9k
12
votes
3
answers
1k
views
Non-commutator in simple group?
Hi,
For a group $G$, we say that $x\in G$ is a commutator if there exists $a,b \in G$ such that $x=a^{1}b^{-1}ab$, and we say that $x$ is a non-commutator if there is no $a,b \in G$ such that $x=a^{1}...
- 2,829