Questions tagged [combinatorial-group-theory]
Use this tag for questions about free groups and presentations of a group by generators and relations.
606
questions
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What group is presented by $\{a,b\mid aaa,aba'bbbab'\}$ and what is its order?
So far I've been able to resolve all finite groups presented by relators with a total length of 11 except this one. I've run Todd-Coxeter until I ran past 256GB of memory; I've run Knuth-Bendix for ...
0
votes
1
answer
53
views
Extending bijection on generators to isomorphism of groups
Given two group presentations $G=\langle X \vert R\rangle$ and $G=\langle Y \vert S\rangle$, let $f:X\to Y$ be a set function only on the generators such that $f(x_1)\ldots f(x_n)=1$ for $x_1\ldots ...
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Show that the group with this presentation has order $12$ and is isomorphic to the alternating group $A_4$ [closed]
Show that the group $$G=\langle a, b, c; a^3, b^2, c^2, ab=ba^2, ac=ca, bc=cb\rangle$$has order $12$ and is isomorphic to the alternating group $A_4$.
I did manage to show that this group is of order ...
0
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0
answers
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Davenport Constant of Symmetric Group S5
I am working on computing the Davenport constant $D(G)$ of symmetric groups, which is the minimal number d such that every sequence of d elements, possibly with repetitions, of a fixed group is one-...
- 19
5
votes
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answers
55
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Product of two iterated commutators and their length.
Given the free group $F_n$ on $n$ generators,
I strongly suspect and want to show that if we are given two iterated commutators $w_1,w_2$ which are not inverses of each other, then the length of their ...
- 131
0
votes
1
answer
75
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Computing whether two finite groups are isomorphic (in C++) [closed]
I need to algorithmically compute whether two given finite groups are isomorphic. Usually I only have generators of these groups.
The groups can get quite large as I'm working with subgroups of $S_{32}...
- 729
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Let $\operatorname{sr}(K)$ be the subgroup rank of $K$. When is $\operatorname{sr}(H)\le\operatorname{sr}(G)$ for $H\le G$?
The Question:
When is $\operatorname{sr}(H)\le\operatorname{sr}(G)$ for (not necessarily abelian) groups $H\le G$ (in general$^{\dagger}$)?
Here $\operatorname{sr}(G)$ is the subgroup rank of a $G$.
...
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For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. Compute subsets of $X$ and $Y$ that generate isomorphic subgroups of $G$.
For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. I need to compute subsets $X' \subset X$ and $Y' \subset Y$ that generate isomorphic subgroups of $G$: $\langle X' \rangle \leq ...
- 729
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vote
0
answers
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Relators in a finitely generated metabelian group
Consider the 2-generated metabelian group $G$ with finite $\mathbf{A}^2$-presentation
$$ \langle\langle a,b \,\vert\, [a,b^{-1}][a^{-1},b][a,b]^2 \rangle\rangle, $$
where $[a,b] = a^{-1}b^{-1}ab$, i.e....
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votes
3
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Proving that a group is infinite and nonabelian
As an exercise I am trying to prove that the group $$G = \langle a,b,c \mid ac = ba, ab=ca, bc=ab\rangle$$ is infinite and non-abelian. Moreover, the author claims that its center has finite index.
I ...
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1
vote
1
answer
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If $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$, Prove that $G/H \cong \mathbb{Z}$
I am trying to prove that if $G := \langle a_k : k \in \mathbb{Z} \rangle$ and $H:= \langle a_{k+1} - a_k : k \in\mathbb{Z} \rangle$ with both of them being free abelian groups, then $G/H \cong \...
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Are amalgams HNN-extensions?
I'm studying geometric groups and graphs of groups, and am trying to build intuition. I'm trying to view amalgams and HNN-extensions in the following way, from the HNN extension Wikipedia:
Thus, HNN ...
- 846
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votes
1
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2 generated groups, with generators of order 3
I can show that if $G$ is a group generated by $a$ and $b$, with $a^2=b^2=1$, then $G$ is a finite elementary abelian 2 group or a dihedral group (finite or infinite), depending whether $ab$ has ...
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Explicitly finding a finite abelian group from a presentation, $\langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$
I have the following group presentation:
$
G= \langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle
$
It is clear that $G$ is a finite abelian group. I am interested in knowing what exactly $G$ ...
- 127
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vote
1
answer
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In a finitely presented group, the set of all words which equal 1 in the group is a recursively enumerable set.
I was reading "An introduction to the theory of groups" by Rotman, chapter 12, the word problem. I am stuck in the following theorem,
Let $G$ be a finitely presented group with presentation $...
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