The combinatorial-group-theor tag has no wiki summary.

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### 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**

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**2**answers

497 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 ...

**4**

votes

**1**answer

123 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

**3**answers

204 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 $SL(2,\mathbb{F}_2)$ ...

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95 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**

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**2**answers

435 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

**1**answer

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

**1**answer

262 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

**1**answer

112 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

**2**answers

265 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 $G$ ...

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144 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 $H \leq \Gamma_g$ ...

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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 ...

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**2**answers

210 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**

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**1**answer

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**

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**1**answer

61 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**

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110 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 ...

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211 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**

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**1**answer

198 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 ...

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**0**answers

1k 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

**1**answer

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

**1**answer

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**

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**2**answers

410 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 ...

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**0**answers

229 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

**1**answer

182 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 ...

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337 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**

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**1**answer

376 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 ...

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**1**answer

574 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 ...

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220 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 ...

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**2**answers

410 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

**1**answer

553 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 ...

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**1**answer

492 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**

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239 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**

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**1**answer

267 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 ...

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541 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**

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**1**answer

263 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**

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**1**answer

242 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**

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**1**answer

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

**1**answer

455 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

**2**answers

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 ...

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**0**answers

208 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

**1**answer

564 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 ...

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votes

**3**answers

514 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 ...

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**1**answer

266 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**

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**1**answer

601 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 ...

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**1**answer

284 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 ...

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**2**answers

248 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 ...

**1**

vote

**1**answer

103 views

### Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...

**1**

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**1**answer

559 views

### Any subgroup of f.g. free group with finite index contains a term of lower central series?

Hello?
I have some questions in the group theory.
I know that the intersection of the lower central series of a finitely generate free group is trivial.
So I wonder whether every nontrivial subgroup ...

**4**

votes

**1**answer

831 views

### For what finite groups is the cardinality of a minimal generating set well defined?

Recently I learned that the cardinality of a minimal set of generators of a finite $p$-group
$G$ is well defined namely it is equal to the dimension of $H^1(G,\mathbb{F}_p)$. Moreover, if
...

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**2**answers

620 views

### infinite group that maps onto dihedral group

The group is generated by $y_i$, $i=0, ...,p-1$
with relations
$y_0y_1=y_1y_2=...=y_{p-1}y_0$
$y_0y_2=y_1y_3=...=y_{p-1}y_1$
$\vdots$
$y_0y_{p-1}=y_1y_0=...y_{p-1}y_{p-2}$
I have run into this ...