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0
votes
0answers
48 views

What is the size of the automorphism group of an abstract polytope with Schläfli symbol $\{p,q\}$?

Let $\Gamma_{p,q}$ be the automorphism group of the aforementioned abstract polytope. What is the size of this group?
0
votes
1answer
48 views

Rank of a generall linear group over a finite field [closed]

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.
2
votes
4answers
160 views

Are the orders of the generators of a group and the product of pairs of thereof enough for this group to be isomorphic to a Coxeter group?

Let's say we have $n$ generators $x_1, x_2, \cdots, x_n$ along with the following facts concerning their orders: \begin{eqnarray*} ord(x_i) &=& 2 \text{ for } i = 1, 2, \cdots, n \\ ord(x_i ...
3
votes
3answers
110 views

Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by \begin{eqnarray*} s_0 &=& a, \\ s_1 &=& b, \text{and} \\ s_{n+2} &=& s_{n+1} ...
13
votes
1answer
442 views

Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper [Brian, Mislove, Every compact group can have a non-measurable subgroup]. A positive solution to a variation of the following problem implies a ...
7
votes
2answers
520 views

Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
5
votes
1answer
131 views

K-fellow traveler property and automatic structure

I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the ...
2
votes
3answers
222 views

Rank of a special linear group over a finite field

What is the rank (minimal number of group generators) of $SL(n,\mathbb{F})$ in the situation when $SL(n,\mathbb{F})$ is not perfect (i.e. when $SL(n,\mathbb{F})$ is different from $SL(2,\mathbb{F}_2)$ ...
1
vote
0answers
99 views

Intuitive meaning of benign subgroup

Disclaimer! This is a copy of a question I posted on M.SE! I still think the question belongs there but I'm not getting any answers so I'm dublicating with slight changes: I've been studying a proof ...
13
votes
2answers
439 views

Why is “The Higman Rope Trick” thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma: If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...
3
votes
1answer
221 views

Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper: Cohen called $A^R_n$ "a standard tool used in ...
6
votes
1answer
275 views

Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
1
vote
1answer
113 views

Request a paper by Fred Cohen

I am looking for the following paper by Cohen, F. R.: On combinatorial group theory in homotopy. Homotopy theory and its applications (Cocoyoc, 1993), 57–63, Contemp. Math., 188, Amer. Math. ...
5
votes
2answers
273 views

Which groups are LERF?

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
2
votes
0answers
147 views

Marshall Hall's theorem for surface groups [closed]

Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$ Let $H \leq \Gamma_g$ ...
0
votes
0answers
55 views

Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein. Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...
4
votes
2answers
213 views

A Karrass-Solitar theorem for surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$ Is there a ...
5
votes
1answer
190 views

Bases of surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
0
votes
1answer
62 views

Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...
4
votes
0answers
111 views

When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
7
votes
2answers
213 views

Modifying Dehn's algorithm to allow equal length replacements?

I'm an analyst trying to understand a certain class of finitely presented groups (one example is below) so it's quite likely this question is naive but I hope it is at least intelligible. Given a ...
7
votes
1answer
202 views

Minimal normally generating subsets of minimal generating sets

Let $G$ be a finitely generated group. The weight $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called ...
3
votes
0answers
2k views

How many combinations does Android pattern have? [closed]

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumps 3) Once a dot is crossed, you can jump over it.
3
votes
1answer
95 views

Long words represent by multiplication of short words

Give a free group $G$ and one of its subgroup $H$ satisfies $rank(G)=n$$[G:H]=k$ Fix a generators of $G$ so we can talk about the length of elements in $G$.Then do there exist constants $A,B,C$ which ...
-1
votes
1answer
149 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
7
votes
2answers
417 views

Number of subgroups of a given index of a free group

Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of rank $n$ have? In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups. I think I ...
8
votes
0answers
238 views

Computing van Kampen diagrams

If G is a finitely presented group (with generating set X) and w is a word over X such that w=1 in G, then the latter can be witnessed by a so called van Kampen diagram for w, which is a planar ...
1
vote
1answer
184 views

how to classify epimorphisms from a subgroup to itself?

Assume $G$,$\hat{G}$ are both free group of rank $n$,and $H$,$\hat{H}$ be their subgroups of index $k$ respectively,$h:H \rightarrow G$, $\hat{h}:\hat{H} \rightarrow\hat{G}$, are two epimorphisms. We ...
8
votes
2answers
338 views

generators of free group

Give a rank $n$ free group $G=\langle a_1,a_2,\dots,a_n\rangle$, let $g_1,g_2,\dots,g_n \in G$, $b_j=g_j^{-1}a_jg_j$ . If $b_1,b_2,\dots,b_n$ can generates the whole $G$, what can we say about ...
7
votes
1answer
395 views

“Concretely” writing down elements in a free profinite group

Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
11
votes
1answer
601 views

Dehn's algorithm for word problem for surface groups

For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
9
votes
0answers
226 views

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 ...
0
votes
2answers
418 views

quotient groups of the lower central series of a free group

I have a question about some quotient groups of the lower central series of a free group. When there's a free group $F = \langle x_1,\cdots, x_n, y_1, \cdots, y_m\rangle $, let $A$ be the subgroup ...
1
vote
1answer
598 views

Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...
17
votes
1answer
504 views

What is the length of the shortest law of $S_n$?

What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$? There is a simple argument showing that we must have $\ell(w)\geq n$. See here for instance. ...
0
votes
0answers
244 views

Finitely presented group and its subgroups

Suppose I have a finitely presented group $G$. By this, I mean I know explicitly what $S$ and $R$ are such that $G = \langle S \mid R \rangle$. Suppose I have a subgroup generated by a finite set of ...
9
votes
1answer
269 views

Does every group embed into a co-hopfian group?

A group $G$ is co-hopfian if every injection $f\colon G \rightarrow G$ is an automorphism, or equivalently if $G$ is not isomorphic to any of its proper subgroups. Miller and Schupp, using small ...
13
votes
2answers
556 views

Kernel of linear representation of Baumslag-Solitar group

Let $BS(m,n)$ be the Baumslag-Solitar group defined by $B(m,n) = < a,b ~|~ b a^m b^{-1} = a^n > $, $mn \neq 0$. There is a linear representation of $BS(m,n)$ by mapping $a$ to the matrix ...
8
votes
1answer
266 views

The equality problem between conjugate group elements

The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursively unsolvable. ...
2
votes
1answer
244 views

Presentations of infinite index subgroups

Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$? If $H$ has finite ...
0
votes
1answer
164 views

Monodromy in presentations of one group over another

Consider a finitely presented group $G$ with presentation $P$ given by $\left\langle g_1,\ldots,g_n|\, r_1,\ldots,r_m\right\rangle$, equipped with a homomorphism $\rho\colon\, G\to H$ to a finitely ...
2
votes
1answer
461 views

Questions on the group with two generators $a,b$ and one relation $b^2=1$

Let $G$ be the finitely presented group with two generators $a,b$ and one relation $b^2=1$. First question: Does that group have a name ? Perhaps an answer to this question can lead me to ...
4
votes
2answers
300 views

Generating a group by randomly sampling generators

Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
5
votes
0answers
209 views

Any method to detect subgroup generated by a subset of the generators from its presentation

I have met the following problem. A group $G$ is given as follows $G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$ Is the subgroup generated by $y$ and $t$ just the ...
3
votes
1answer
567 views

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one ...
5
votes
3answers
529 views

Results in the Presentation of Finite Groups

I've been looking at combinatorial group theory, but all the results seem to be about infinite groups. Are there any important results about the presentations finite groups specifically (or are useful ...
8
votes
1answer
267 views

question about derived subgroup

Let $G$ be a free group. Then $G/G^{(n)}$ ($G^{(n)}$ is the $n$th derived subgroup.) acts on $G^{(n)}/G^{(n+1)}$ by conjugation, which makes $G^{(n)}/G^{(n+1)}$ a $\mathbb{Z}[G/G^{(n)}]$-module. What ...
2
votes
1answer
618 views

Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups: 1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$, 2) With orders that are ...
1
vote
1answer
285 views

When $[G_k,G_m] = G_{k+m}$?

Hello? I have a simple question about combinatorial group theory. For a group $G$, I saw $[G_k, G_m] \subset G_{k+m}$ and these two subgroups need not be equal. Then is there any known condition that ...
3
votes
2answers
249 views

In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?

For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M ) then we use the ...