# Questions tagged [group-presentation]

For questions concerning groups defined via a presentation by generators and relations.

445 questions
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### What is a simply presented group?

I have some background in commutative ring theory. At the moment I am going through factorization theory of integral domains. I found out that it is a conjecture, that every Abelian group is the ...
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### Understand the free group universal property applied to $D_n$

For $n ≥ 3$ and $D_n$ the dihedral group of order $2n$ with présentation $\langle r, s : r^n = s^2 = srsr = 1\rangle$ prove that for all $(a, b) \in (\Bbb Z/n\Bbb Z)^2$, there exists a morphism $f$ ...
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### How to classify the sets $M$ by their structures?

In this post, we construct a set of matrices with the following properties Given $M$ comprised of $n\times n$ matrices, which satisfies $I_n \in M$ and $0_{n} \not\in M$ If $A,B \in M$, ...
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### How can I show that $D_{2n} \cong C_n \rtimes C_2$

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
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### Presentation of a group generated by reflections through hyperplane

Question: Let $P_i\in\mathbb{R}^n$ be the hyperplane $x_i - x_{i+1} = 0$. Find a presentation for the group $G$ generated by the reflections in $P_1, \ldots, P_{n-1}$ Attempt: I really don't know how ...
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### Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (...
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### Find the isomorphism type

Consider the abelian group $G$ generated by $a$, $b$ and $c$ and determined by the following relations \begin{aligned} 3 a+9 b+9 c &=0 \\-3 b+9 c &=0 \end{aligned} determine the isomorphism ...
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### Proving the Free Abelian Group is Free Abelian…?

On page 40 of these notes is the following exercise: Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$. On ...
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### Can I conclude that my group is finitely generated, if it is a homomorphic image of a free-group on finitely many generators?

Say $X$ is a finite set, $F \langle X \rangle$ is the free group on the set $X$ and $G$ be a group. If I have a surjective homomorphism $$\varphi : F \langle X \rangle\longrightarrow G$$ then can I ...
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### Quaternion Group: Determine that $i^4 = 1$.

Suppose we are given the following presentation of the quaternion group: $Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$ Is it obvious that $i^4 = 1$?
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### What is an algorithm for determining if a finitely presented group is finite

Suppose I am given a presentation of a group with a finite number of generators and a finite number of relations. Is there an algorithm for determining if the group is finite? Also, if there is such ...
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### Could $\langle \Gamma | R \rangle \cong \langle \Gamma | S\rangle$ if $\langle R\rangle \subsetneq \langle S\rangle$?

If we have two finitely presented groups $\langle \Gamma | R\rangle$ and $\langle \Gamma | S\rangle$ with $\langle R\rangle \subsetneq \langle S\rangle$, could they be isomorphic?
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### How do I turn a group presentation into a multiplication table?

Suppose I have a group presentation and I know for a fact that this is a presentation of some finite group. How do I create a multiplication table from this presentation?
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### Deriving the Discrete Heisenberg Group generators.

How can we derive the generators of the Discrete Heisnberg Group? Everyone seems to just state this as a given and never actually derive it from scratch. I'm looking for a (somewhat) elementary ...