Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,941
questions
22
votes
1
answer
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Monstrous moonshine for $M_{24}$ and K3?
An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...
1
vote
1
answer
80
views
Complexity to decide for permutation group if every element fixed at most $k$ points
I want to consider the following problem, which generalises the decision problem to decide if a given finite permutation group is a Frobenius group:
Given a finite permutation group in terms of its ...
1
vote
2
answers
190
views
Complexity of decision problem to decide if permutation group is $k$-transitive
Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a ...
0
votes
0
answers
136
views
Finite group and cyclic cover
Suppose the finite group $N$ surjects to finite group
$F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and
surject to $F$.
But is this ...
3
votes
1
answer
206
views
Action of homeomorphism on real line
An element $f ∈ Homeo^+(\mathbb{R})$ is said to be of :
(1) type A, if it has a trivial germ at ∞ and it does not fix any point in an
interval of the form (−∞, r).
(2) type B, if it has a trivial ...
3
votes
1
answer
110
views
How many length-24 Type III codes have no words of Hamming weight 3?
From W. Cary Huffman (2005), On the classification and enumeration of self-dual codes, Finite Fields and Their Applications
11(3) pp 451-490, I learn that there are at least 140 Type III codes of ...
10
votes
2
answers
682
views
Order of unipotent matrices over $\mathbb{Z}/q\mathbb{Z}$
Let $q$ be a prime power and let $n\geq2$ be an integer.
Is it known what is the largest order of a unipotent upper-triangular $n\times n$ matrix over the ring $\mathbb{Z}/q\mathbb{Z}$?
I am mostly ...
41
votes
1
answer
3k
views
Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup?
Edit on March 2, 2018: I just noticed that this is almost identical to a question asked on MO by David Harden in 2011, and that I had even given an (incomplete) answer to that one. I would delete the ...
11
votes
0
answers
174
views
Iterated automorphism groups of finite groups
Let $\mathcal{G}$ be the set of isomorphism classes of finite groups.
There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, ...
-1
votes
1
answer
223
views
Group isomorphism $R^{\times} \simeq C_n\times C_2$? [closed]
Let $n$ be an positive integer and $R:=\mathbb{Z}[X]/(X^2,nX)$.
Do we have $R^{\times} \simeq C_n\times C_2$ (the group of units of $R$) as it seems to be the case for small values of $n$.
If so, do ...
6
votes
1
answer
337
views
Relation between commutator length and stable commutator length in free groups
In Bardakov, Algebra and Logic, Vol. 39, No. 4, 2000 I have found the following (page 225, see https://link.springer.com/article/10.1007/BF02681648)
We pronounce tile validity of the following:
...
6
votes
1
answer
893
views
distortion of cyclic subgroups of linear groups
In an informal talk I heard a statement:
"Any cyclic subgroup in a linear group is at most exponentially distorted"
with a vague reference to a work of Lubotzky with coauthors.
The works of ...
5
votes
2
answers
249
views
Braid groups on topological spaces
The configuration space $C_n(M)$ of $n$ particles in some connected graph $M$ (thought of as the topological realisation of a one-dimensional CW-complex) is
$$M^n \backslash \{ (x_1, \ldots, x_n) \...
7
votes
3
answers
798
views
Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II
For some context see Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
As per Noah's answer and JBL's comment this was false as stated. However, I think the following ...
2
votes
1
answer
73
views
Maximal subgroups of some special cases of ${\rm L}_{n}^{\epsilon}(q)$
Let $G$ be a finite simple group of type ${\rm L}_{n}^{\epsilon}(q)$ with the following conditions:
$n$ and $\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ both prime, $n\geqslant3$ and $(n,q,\...
3
votes
2
answers
183
views
Presentations of superperfect groups
Are there non-trivial superperfect groups with the property that there exists a presentation of the group where the number of generators equals the number of relations?
5
votes
0
answers
207
views
Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
5
votes
2
answers
242
views
Counting transitive generators according to coset type
Let $\sigma=(1\;2)(3\;4)\cdots (n-1\; n)$ be a fixed-point-free involution in $S_{2n}$. I want to count permutations $\pi$ such that the group $\langle \pi,\sigma\rangle$ generated by $\pi$ and $\...
7
votes
0
answers
186
views
Elements of a group than can achieve all orders in its quotients
Let $G$ be a group. For different purposes I am now interested in a quite natural property that the elements in $G$ may or may not have, and I would like to ask if there is a standard terminology for ...
2
votes
1
answer
143
views
Every quasicharacter of an open subgroup extends to a quasicharacter on the whole group
Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous ...
5
votes
2
answers
448
views
Is each locally compact group topology on the permutation group discrete?
Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete?
Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
14
votes
4
answers
669
views
Non-split Aut(G) $\to$ Out(G)?
I'm looking for examples of outer automorphisms of a finite group $G$ which do not lift to automorphisms (i.e. non-split quotient map $\mathrm{Aut}(G)\to \mathrm{Out}(G)$, where $\mathrm{Out}(G) = \...
5
votes
1
answer
298
views
Amenability of $S^{\infty}$
Let $G$ be the group of all permutations of $\mathbb{N}$. If I am not mistaken, this group is denoted by $S^{\infty}$.
Is there a precise locally compact topology on $G$ such that $G$ would ...
4
votes
1
answer
346
views
A generalization of Siegel property
In reduction theory of arithmetic groups, one has the following finiteness property.
Proposition 1 (Siegel property). Let $G$ be a reductive group over $\mathbb{Q}$ and let $\Gamma\subset G(\mathbb{...
114
votes
3
answers
5k
views
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
2
votes
0
answers
202
views
Random walk on a finite group, converging modulo a function
Let $G$ be a finite group, and let $Q$ be a probability measure on $G$. Suppose that $Q$, as a function on $G$, is supported on a conjugacy class $C$. We denote by $Q^{*k}$ the $k$-fold convolution of ...
4
votes
2
answers
488
views
Co-finite type abelian groups
Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite.
Let $B_{\rm tor} = \varinjlim_{n\ge 1} B[n]$ be the torsion subgroup of $B$.
Is it ...
15
votes
1
answer
597
views
What is this quotient of the triangle 2-3-7 group?
I have been working with Hurwitz groups, and I came across the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, ([a,b]^2ab)^6 \rangle$. I'm trying to figure out exactly what this group is. I know it ...
3
votes
0
answers
256
views
A question about continuous group cohomology
Let $G$ be a profinite topological group, $M$ a discrete $G$-module.
If $M$ is "P", is every $H^i_{\rm cont}(G,M)$ also "P"? or at least is it a subgroup/subquotient of an abelian group that is "P"? ...
8
votes
0
answers
414
views
Is the class of commutative generalized Euclidean rings stable under quotient and localization?
Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
5
votes
3
answers
372
views
Graphs of groups with homomorphisms not necessarily injective
I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general ...
2
votes
2
answers
151
views
How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]
How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...
7
votes
0
answers
306
views
On an inequality concerning the strict cohomological dimension of a profinite group
This is an exercise from Serre’s book on Galois cohomology.
Let $G$ be a profinite group and $H$ a normal closed subgroup and suppose that the cohomological dimension at the prime $p$ of $G/H$ is ...
1
vote
0
answers
109
views
Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
4
votes
1
answer
223
views
No lifts in an exact sequence of profinite groups?
In pg. 24 of his book on Galois cohomology, Serre gives the following exercise:
"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:
(i) $...
4
votes
2
answers
353
views
Random walk uniformly hitting a compact set
Suppose $G$ is a locally compact compactly generated group. Let $\mu$ be a probability measure that is:
Adapted to $G$, i.e. there is no proper subgroup $H$ such that $\mu(H)=1$.
Symmetric, i.e. $\...
10
votes
0
answers
209
views
2-generator subgroups of an Artin group of small type
Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...
15
votes
2
answers
910
views
Subgroup of hyperbolic group generated by non-torsion elements
Let $G$ be a hyperbolic group. I know that it is an open problem whether $G$ has a torsion-free subgroup of finite index. But if we let $N$ be the subgroup of $G$ generated by its non-torsion elements,...
11
votes
1
answer
159
views
A group of type F that is an extension of type F-by-type F
Let us first recall that a group of type $F$ is a group admitting a compact classifying space.
Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
0
votes
1
answer
315
views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
9
votes
3
answers
687
views
Reduction mod $n$ of symplectic group
Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–...
1
vote
0
answers
42
views
Alternating Hurwitz quotients multiplicity
How many times is each alternating group a hurwitz quotient? In other words, how many non-isomorphic ways can an alternating group be generated by an element of order 2 and an element of order 3 whose ...
3
votes
0
answers
688
views
Union of the conjugates of maximal subgroups
This post is a generalization of Union of the conjugates of a proper subgroup.
Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that:
(1) $ \...
2
votes
1
answer
242
views
Even Isometries in neutral Geometry
Consider a Hilbert plane as in Hartshorne's 'Euclid and beyond' (axiomatic geometry), and its group of isometries f or 'rigid motion' generated by line reflections. Call f 'even' if it is the product ...
14
votes
1
answer
885
views
The number of involutions in a permutation group
If $G$ is a group let $I(G)$ be the number of involutions (elements of order 2) in $G$. My question is then easily stated: does there exists a constant $C > 1$ such that for every $n \ge 1$ and ...
2
votes
0
answers
79
views
Is there any nice way to compute transfer homomorphism in a $p$-group?
Let $G$ be a $p$-group and $H$ be a subgroup of $G$. Set $V$ be a pretransfer map from $G$ to $H$.
That is, $$V(g)=\prod_{t\in T} tg(t.g)^{-1}=\prod_{t\in T_0}tg^n_tt^{-1} $$
Can we say that the ...
2
votes
0
answers
66
views
What are the transitive extensions of finite representations of cyclic groups?
This question is a generalisation of this one. Let $H$ be a finite, transitive permutation group of degree $n$. If the point stabiliser subgroup $H_n$ of degree $n-1$ is some faithful permutation ...
4
votes
0
answers
224
views
Compactifications of reductive groups via representation theory
Let $G$ be a reductive group, $\Lambda$ a weight lattice, $\Lambda^{+}$ the monoid of dominating weights, $\omega_1,\dots,\omega_r\in \Lambda^{+}$ the fundamental weights and $\{\alpha_1,\dots, \...
1
vote
0
answers
65
views
Is there a decomposition exists for $e^{c(K_++K_-)^2}$
In the usual $SU(1,1)$ group:
$$[K_+,K_-]=-2K_z,\quad [K_z,K_\pm]=\pm K_\pm.$$
Is there a decomposition exist for $e^{c(K_++K_-)^2}$?
Of course there won't exist a decomposition to $e^{K_+},e^{K_-},...
3
votes
1
answer
216
views
Intersections of products of Sylow $p$-subgroups
Motivated by the work of Cohn, Umans and collaborators on bounding the complexity of matrix multiplication, a student and I have been investigating the following problem.
For subsets $X$ and $Y$ of a ...