# Questions tagged [group-presentation]

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

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### Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle$$ is equivalent to the trivial group. I have tried all ...
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### Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
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### Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? [duplicate]

Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$? I've just been playing around with the relations but I can't seem to get ...
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### Why the group $\langle x,y\mid x^2=y^2\rangle$ is not free?

Why is the group $G= \langle x,y\mid x^2=y^2\rangle$ not free? I can't find any reason like an element of finite order or some subgroup of it that is not free etc.
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### Does this group presentation define a nontrivial group?

Given a presentation $$\langle x,y,z : x^y=x^2, y^z=y^2, z^x = z^2 \rangle,$$ where $x^y$ is just the usual conjugation. Can we say for sure, whether this presentation defines a nontrivial group?
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### Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.

I'm trying to prove that the following group has infinite order: $$H=\langle a,b\mid a^{3}=b^{3}=(ab)^{3}=1\rangle.$$ Currently I'm checking on some cases using the relations, but my problem is the ...
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### Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle$$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
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### Finitely Presented is Preserved by Extension

Given $N= \langle n_i|r_j \rangle$ and $G/N= \langle g_k|s_l \rangle$, how do we prove $G$ has a finite presentation? We know that $G$ is f.g. by $\{n_i,g_k\}$ (I am being sloppy about directly ...
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### The order of a group presentation

Find the order of the group $G$ which has the presentation $\langle a,b \mid a^{16}=b^6=1,bab^{-1}=a^3\rangle$ I found that $a^8b=ba^8$ hence $\langle a^8,b\rangle$ is an abelian sungroup of $G$. ...
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### groups of order $p^4$

I need the classification of finite non-abelian groups of order $p^4$ from E. Schenkman's book "Group theory, 1965". Unfortunately our library has no this book and there does not exist the full ...
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### Is a HNN extension of a virtually torsion-free group virtually torsion-free?

Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free group. Let $H,K<G$ be isomorphic (finite index) subgroups of $G$ and let $\varphi:H\rightarrow K$ be an isomorphism. ...
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### If I have the presentation of a group, how can I find the commutator subgroup of it?

I have the group given by the presentation $G= \langle a,b\mid a^2,b^2\rangle$ How can I in general find $G',G/G',G''$ ? thanks for any hints.
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### Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?

All is in the title: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$? I think it is the case, but I don't know how to prove it.
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### Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
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### Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can't find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the ...
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### Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
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### Finding the index of $N$ in $F$.

This is Exercise 2.1.4(a) of "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by W. Magnus et al. The Question: Let $F=\langle a, b\rangle$. If $N$ is ...
### Identifying $\langle x_e, x_a, x_b, x_{ab}\mid x_ex_a=x_{ab}, x_ax_e=x_b, x_bx_{ab}=x_a, x_{ab}x_b=x_e\rangle.$
Consider the group presentation $$\langle x_e, x_a, x_b, x_{ab}\mid x_ex_a\stackrel{(1)}=x_{ab}, x_ax_e\stackrel{(2)}=x_b, x_bx_{ab}\stackrel{(3)}=x_a, x_{ab}x_b\stackrel{(4)}=x_e\rangle.$$ What group ...
### Show that the class of the word $aba$ in group $\langle{a,b\,\vert\,a^{2}b^{2} }\rangle$ is not the trivial element. [closed]
Could anyone give me a suggestion to solve this problem about group presentations? Show that the class of the word $aba$ in group $\langle{a,b\mid a^{2}b^{2} }\rangle$ is not the trivial element.