# Questions tagged [group-presentation]

For questions concerning groups defined via a presentation by generators and relations. Should probably be used along with the general (group-theory) tag.

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### Is the group $\langle x,y \mid xy^n = yx^n \rangle$ trivial?

I’m currently browsing through Clara Löh’s book “Geometric Group Theory - an introduction”, and came across the exercise 2.E.14, which gives some examples of presentations of groups and asks to ...
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### Can the following proof calculus show that any finitely presented free group is free?

If a finitely presented group is free, will it always have a proof in the proof calculus outlined in this question that it is free? I recently saw this question. I tried to show that the group was ...
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### Dehn presentation implies finitely many conjugacy classes of elements of finite order

Let $G$ be a finitely generated hyperbolic group. Show that $G$ contains only finitely many conjugacy classes of elements of finite order. In “Geometric Group Theory: An Introduction” by Clara Löh, it ...
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### Presentation of Product Group

Here is the question I have been working on: If $G_1 = \langle X_1 : R_1\rangle$ and $G_2 = \langle X_2 : R_2\rangle$, supply a presentation for $G_1 \times G_2$. Deduce that, if $G_1$ and $G_2$ are ...
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### Can you completely determine a finitely presented finite group?

Let $G = \langle S \mid R \rangle$ be a finitely presented group. Suppose you know $G$ is finite. Can you completely construct the group multiplication table? I feel like the answer is yes. My first ...
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### Trying to find the set of unique representatives for the geodesics in the group $\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$

I am studying the conjugacy growth of the groups, and I encountered the following group: $$G=\langle a,t \mid ata^{-2}t^2a^{-2}tat^{-4}\rangle$$ Thanks to Derek for pointing out that $G$ is an ...
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### presentation of a family of solvable groups

I am looking for a family of finite solvable groups (with infinite members) with given presentations. I am aware that for example dihedral groups, or dicyclic groups have well-known presentations. ...
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### $\text{GL }_2(\mathbb{Z})$ as an HNN extension

This question arises from section $2$, exercise $13.8$ of Bogopolski's Introduction to Group Theory. I managed to show that $\text{GL}_2(\mathbb{Z})\cong D_4 *_{D_2} D_6$. Now, I want to take a random ...
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### Number of groups with a bounded short presentation

How many groups there are (up to isomorphism) with a presentation with at most $n$ generators and with relators of length at most $3$? I don't expect there exist a sharp solution, since I know that ...
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### Writing the product of cycles as disjoint cycles, but yielding to different result when applying it.

I'm studying the symmetric group $S_n$ of permutations from $\{1,2,\ldots,n\}$ to itself. I'm doing a problem where I have to show that $$S_4\cong \langle a,b~|~a^4=b^2=(ba)^3=1\rangle.$$ I thought ...
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### Is the isomorphism problem solvable for Euclidean groups?

Suppose you had two group presentations, and you know they are Euclidean groups, can you tell if they are isomorphic or not? It has been suggested to me that it is probably possible to tell if they ...
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### Word problem for groups and the universal cover

I am confused by certain constructions regarding fundamental groups. The word problem for finitely presented groups is known to be undecidable. However, it is also known how to construct, given a ...
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### Is this a valid group structure (of order 12)?

Let $$G = \big\langle \, a, b, c \colon a^3 = b^2 = c^2 = (ab)^2 = (bc)^2 = 1, ac = ca \big\rangle. \tag{0}$$ That is, $G$ is a group that has elements $a$, $b$, and $c$ (though these are not the ...
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### Prove that $S_4$ is isomorphic to a presentation

I would like to prove that $G=\langle a,b \, | \, a^2,b^4,(ab)^3\rangle \cong S_4$. I tried to list out all the elements in the group presentation and show that it is isomorphic to $S_4$, but it was ...
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### Identify the presentation of groups [closed]

I'm trying to classify all groups of order $20$. I get two group presentation and want to identify isomorphism type. I think one of these isomorphic to $D_{10}$ and other isomorphic to $Dic_5$. Can ...
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### Show that $G = \langle a,b \mid a^2b = ba^3\rangle$ is nonabelian.

I was trying to suppose for a contradiction that it is abelian, then: $$ba^3 = a^2b = ba^2\implies a = 1$$ Then we have $G = \langle a,b\mid a = 1\rangle$, which I believe is isomorphic to $\mathbb{Z}$...