Questions tagged [group-presentation]

Question 1

I am currently taking a course on Free groups and we have the following proposition from B. Neumann, 1937: if $ G = ⟨x_1,...,x_n|r_1,...,r_m⟩ = ⟨y_1,...,y_k|S⟩$, then there exists a finite subset $S_0=...

Question 2

Let $G$ be a group with presentation $\langle S \ \vert \ R \rangle$. Then any $g \in G$ has (in case $R$ not trivial in general non unique) the form $g=s_1^{a_1}s_2^{a_2}...s_n^{a_n}$ with $s_i \in S$...

Question 3

Let $G$ be a group with presentation $\langle S \ \vert \ R \rangle$. Then any $g \in G$ has (in $R$ not trivial in general non unique) the form $g=s_1^{a_1}s_2^{a_2}...s_n^{a_n}$ with $s_i \in S$ and ...

Question 4

The Todd-Coxeter algorithm is a useful tool to study groups defined by generators and relations. It computes the action of the group on cosets, giving a homomorphism of the group to the group of ...

Question 5

Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers and let ${\rm SL}_{2}(\mathbb{Z}_{p})$ be the two dimensional special linear group over $\mathbb{Z}_{p}$. Note that ${\rm SL}_{2}(\mathbb{Z}_{p})$ ...

Question 6

I would like to know if there is some literature (or low hanging results) on the construction below, or related constructions for that matter. I'm interested about what can be said in each direction (...

Question 7

Consider the group $G = \langle a, b : a^4 = 1, a^2 = b^2, bab^{-1} = a^{-1}\rangle$. This is a standard presentation of the quaternion group. But why could it not be the Klein four group? My ...

Question 8

Let's say $S_3=\langle x,y \mid x^2,y^3,xyxy \rangle =\langle A\mid R \rangle$. Let $X$ be the subgroup of $F(A)$ that is generated by the element in $R$. $\phi$ is a homomorphism from $F(A)$ to $S_3$....

Question 9

In the group of isometries of the plane, let $r$ be a rotation by $\frac{2\pi}5$ radians and let $s$ be a translation. I'd like to find a finite presentation of the subgroup $G=\langle r,s\rangle$. ...

Question 10

It is well known that the symmetric group $S_n$ ($n\geq 3$) is $2$-generated. For example, one may take as generators $(1\ 2)$ and $(1\ 2\ \ldots \ n)$. This allows us to write every element of $S_n$ ...

Question 11

I'm trying to understand this statement. The free group $F_2=\langle a,b\rangle$ is freely generated, so by the definition there is a homomorphism from $F_2$ to $\mathbb{Z}\oplus\mathbb{Z}=\{c^m,d^n\...

Question 12

I am looking for a reference that proves that $SL(2,\mathbb{Z}) = \langle x,y | x^4 , (xy)^3 = x^2 \rangle$. It sounds like a basic question, but I did not find a proof of this claim. It is mentioned ...

Question 13

A group $G$ is said to be algebraically fibred if there exists an epimorphism $G\to\mathbb{Z}$ with finitely generated kernel. I want to study wheter $G=\langle a,b\mid a^2=1,abab=baba\rangle$ fibres ...

Question 14

The free product $ \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$ contains free nonabelian subgroups. Let's see presentation of a groups: $ \mathbb{Z}/2 = \langle a \mid a^2 = e \rangle$ $ \mathbb{...

Question 15

Using Tietze transformations show that the group $G = \langle x, y, z \mid (xy)^2xy^2\rangle$ is a free group of rank 2. It means that $G = \langle \left \{ w_1, w_2 \right \} \mid \varnothing \...

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Questions tagged [group-presentation]

Understanding why the existence of an isomorphy of a finitely presented group to another one implies the relators is a finite set

Property of Length Function for Presentation of Group II & Case of Coxeter Group

Property of Length Function for Presentation of a Group

Left vs right action in the Todd-Coxeter algorithm

A profinite presentation of ${\rm SL}_{2}(\mathbb{Z}_{p})$

What is known in general about these groups coming from a graph?

Why is this presentation of quaternions not the Klein four group?

I do not understand finitely presented group

Finding a presentation of the group generated by a translation and a rotation

What's the shortest way to represent the elements of $S_n$ in terms of two generators?

$\langle a,b \mid aba^{-1}b^{-1}\rangle$: a presentation of $\mathbb{Z}\oplus\mathbb{Z}$ [duplicate]

Presentation of $SL(2,\mathbb{Z})$

Algebraic fibring of $\langle a,b\mid a^2=1,abab=baba\rangle$.

Why free product $ \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$ contains free nonabelian subgroups? [duplicate]

Tietze Transformations: Show that $\langle x, y, z \mid (xy)^2xy^2\rangle$ is a free group of rank 2 [closed]

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