Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,941
questions
4
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1
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Second cohomology of the adjoint representation
Let $p$ be a prime and let $M_p$ be the $\mathrm{GL}_2(\mathbb{F}_p)$-module of $2 \times 2$ matrices over $\mathbb{F}_p$ with trace $0$ (the action is by conjugation).
Is it true that for $p$ large ...
- 11.2k
2
votes
1
answer
120
views
Generalizing a codistributive property of sufficiently disjoint normal subgroups to protomodular categories
In a poset, whenever the meets and joins below exist, their universal properties induce a containment $$(A\vee B)\wedge (A\vee C)\geq A\vee(B\wedge C).$$ This is an instance of codistributivity. In a ...
- 10.3k
7
votes
1
answer
588
views
Does a central extension split over some finite index subgroup?
Consider a central extension of groups $\mathcal{E}$ :
$$1\rightarrow A\rightarrow \widetilde{G}\rightarrow G \rightarrow 1$$with $A$ finite cyclic. Does there always exist a finite index subgroup $H$ ...
- 37.3k
10
votes
3
answers
2k
views
Web interface for GAP (or other computer algebra system dealing with finite groups)?
GAP is computer algebra system which allows to make calculations with finite groups. (See wikipedia link for an example).
Is there web interface for it ? (I cannot google it.)
Or may be some other ...
6
votes
2
answers
483
views
In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$?
Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$.
Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z}...
- 153
5
votes
1
answer
163
views
Goldman Lie algebra of a bordered surface vs. a closed surface?
How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) ...
- 1,971
21
votes
2
answers
603
views
Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
- 2,329
6
votes
0
answers
407
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Need help remembering elegant but forgotten proof of a forgotten theorem [closed]
I would like to apologize in advance for this question, it's very vague.
I once attended a lecture in Abstract Algebra where the professor proved a surprisingly fundamental theorem in a remarkably ...
- 213
1
vote
0
answers
279
views
A question about Mostow's theorem for self-adjoint groups
In Mostow's paper Self-adjoint groups, one can find the following property.
Theorem. Let $G\subset \mathrm{GL}_{n,\mathbb{R}}$ be a reductive real algebraic subgroup. Then there exists $a\in \...
- 199
3
votes
3
answers
373
views
Group action on R^n [closed]
I am studying the symmetries of a particular function,
$$
f: R^n \rightarrow R
$$
which leave $f(x)$ unchanged (i.e. so $f(Ax) = f(x)$ for some matrix $A \in R^{n \times n}$). I have found that my ...
- 69
17
votes
0
answers
679
views
Monstrous Langlands-McKay or what is bijection between conjugacy classes and irreducible representation for sporadic simple groups?
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group. Moreover for the symmetric group and some other groups there is "good ...
21
votes
1
answer
765
views
Girth of the symmetric group
Let $n \in \mathbb N$ and $\{\sigma,\tau\} \subset {\rm Sym}(n)$ be a generating set.
Question: What is the maximal possible girth (if one varies $\sigma, \tau$) of the associated Cayley graph?
I ...
- 25.3k
11
votes
3
answers
619
views
$SL_n(\mathbb{Z})$ is co-hopfian
Is it true that $SL_n(\mathbb{Z})$ is co-hopfian for $n \geq 3$? I heard about this result, but I can't find any reference. Would be grateful for any references!
- 631
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votes
1
answer
2k
views
What finite simple groups we can obtain using octonions?
Rearranged on 2017-05-31
What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define.
Many of the finite groups are defined using machinery ...
user21230
2
votes
0
answers
139
views
Concentration of Reduced words
This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...
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3
votes
0
answers
183
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 1,987
2
votes
0
answers
95
views
Semidirect product of $G$-posets
Let $P$ be a poset on which a partially ordered group $G$ acts by monotone bijections. I call these guys $G$-posets. If I'm not wrong, the semidirect product of two partially ordered groups remains ...
- 13.2k
5
votes
2
answers
500
views
Finiteness properties of mapping class groups
Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:
1) Does every finite generating set give us a finite ...
- 858
11
votes
2
answers
546
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 ...
- 18.1k
5
votes
2
answers
238
views
Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
- 6,933
1
vote
0
answers
99
views
Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
- 1,651
9
votes
1
answer
870
views
Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
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4
votes
1
answer
317
views
For which groups is (non-)left orderability decidable?
Mainly, my question is in the title, but let me be more precise here.
Let $G$ be a finitely presented group with solvable word problem. If G is not left-orderable, is there an finite-time algorithm ...
- 5,231
1
vote
0
answers
586
views
Inverse limits and first isomorphism theorem for compact topological groups
This question was originally asked on MathSE here.
I have a problem with Proposition (1.2.1) from J. Wilson's book 'Profinite Groups'
The proposition is the following:
Let $(G, \varphi_i : G \to ...
- 41
3
votes
2
answers
169
views
Weak Pronormality of a finite group
Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.
Definition: A subgroup $H$ of $G$ is said to be ...
- 366
11
votes
2
answers
932
views
Subtle point in definition of BNS invariant
Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling ...
- 43.6k
5
votes
0
answers
433
views
Subgroups and quotients of an abelian pro-finite group
It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...
- 93
6
votes
1
answer
285
views
Reference request: Reduced reflection length in Coxeter groups
I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
- 809
4
votes
1
answer
416
views
Finite index subgroups of a RAAG
Let $G$ be the group given by the presentation
$$\langle x,y,z,w \ | \ xy = yx, yz = zy, zw = wz\rangle.$$
This is a right-angled Artin group (RAAG) whose graph is a path on $4$ vertices.
We can ...
- 11.2k
3
votes
0
answers
91
views
Visualization (even locally) of graphs with given infinite group
I want to give a lecture about Frucht theorem (and its generalization) which state that: for each finite group $G$, there is at least one finite graph $\Gamma$ such that $Aut(\Gamma)\cong G$. For each ...
- 4,748
12
votes
1
answer
560
views
Can a simple group be equivalent to a non-simple group?
Two abstract groups $G$ and $H$ are called equivalent, $G\sim H$, if each of them is isomorphic to a subgroup of another.
Question: Can a simple group $G$ be equivalent to a non-simple group $H$?
Of ...
- 1,939
11
votes
2
answers
550
views
Homeomorphisms vs Borel automorphisms
Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...
- 1,939
10
votes
3
answers
475
views
From Gassmann-Sunada triples to isospectral manifolds
A Gassmann-Sunada triple is a triple $(U,V,W)$ of groups, with $V, W$ subgroups of $U$, such that $U$ and $V$ meet every conjugacy class in $U$ in the same number of elements, and such that $V$ and $W$...
- 4,353
6
votes
0
answers
220
views
Lower bound for order of matrix modulo $n$
For a positive integer $n$ and a square matrix $A$ with integral entries, let $\text{ord}(A, n)$ be the smallest positive integer $k$ such that $A^k \equiv \mathbf{1} \bmod n$, if such integer $k$ ...
user40023
2
votes
2
answers
659
views
Subgroup of a free group that is characteristic but not totally characteristic
Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...
- 183
-2
votes
1
answer
506
views
no classification of nilpotent lie groups
there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\...
- 22.6k
7
votes
1
answer
457
views
The group of $k$-automorphisms of $k[x_1,\ldots,x_n,x_1^{-1}]$
Let $k$ be a field (of characteristic zero).
For $k[x_1,\dotsc,x_n]$ it is known that the affine and triangular automorphisms generate $G_n$, the group of automorphisms of $k[x_1,\dotsc,x_n]$,
see, ...
- 2,783
2
votes
2
answers
944
views
are all simply connected nilpotent lie groups matrix groups over $\mathbb{R}$?
Mathworld just says that the lower central series terminates: $\mathfrak{g}_1= [ \mathfrak{g}, \mathfrak{g}]$,
$\mathfrak{g}_2= [ \mathfrak{g}, \mathfrak{g}_1]$ and $\mathfrak{g}_n= [ \mathfrak{g}, \...
- 22.6k
1
vote
0
answers
192
views
Non-existence of nontrivial finite group extension of any simply-connected Lie group
Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that
there does not exist any group $G$ (with no topology) ...
- 10.3k
4
votes
1
answer
332
views
Is there a topologizable group admitting only Raikov-complete group topologies?
Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is ...
- 40.9k
5
votes
1
answer
329
views
Short proof a monoid is a group iff every splitting is right homogeneous
In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...
- 10.3k
5
votes
1
answer
428
views
Identify one group of linear transformations
Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Q})$ defined as follows:
Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Q}^9$, $G$ is generated by:
All ...
- 190
5
votes
2
answers
473
views
Wielandt automorphism tower theorem
I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939).
The theorem states the following:
Given a ...
- 461
2
votes
0
answers
63
views
Determining subgroup of finite group of Lie type from characteristic polynomials
Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
- 2,419
1
vote
2
answers
311
views
group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$
I am trying to understand Kloosterman sums and their estimates (e.g. from [1], which does not prove)
$$ \Big| S(m,n;c) \Big| = \Big| \sum_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, ...
- 22.6k
1
vote
1
answer
179
views
The order of the system normalizer in a finite solvable group
Definition: Let $G$ be a finite solvable group and $\Sigma \in \text{H}(G)$, the set of Hall systems of $G$. The normaliser of $\Sigma$ is defined as $$ N_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for ...
- 366
2
votes
1
answer
196
views
p-group as a product of two abelian normal subgroups
Thanks for any comment or answer.
Let $G$ be a finite non-abelian $p$-group such that $G=AB$ where $A=C_G(a)$ and $B=C_G(b)$ are maximal abelian normal subgroups of $G$ such that $A\cap B=Z(G)$, ...
- 99
2
votes
0
answers
593
views
Volume of $SL(2,\mathbb{C})$ [closed]
So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf
I can write the Haar measure of $SL(2,\mathbb{C})$ as
$$d\mu = \sinh^2(r) dr dk dk'$$
where $r$ runs over nonnegative real numbers ...
- 151
6
votes
0
answers
454
views
On the average number of subgroups per conjugacy class
At some point in the future, I hope to do some work on estimates for the number of conjugacy classes of subgroups of a finite group (by an estimate here, I mean an upper bound). Assuming, for the ...
- 347
14
votes
1
answer
677
views
$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
- 151