# 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 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 $\mathbb{Z}$...
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### Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$.

The Problem: Suppose $G=\left\langle x, y, t\mid x^7=y^7=t^3=1, txt^{-1}=x^2, tyt^{-1}=y\right\rangle$. Show that $y\in Z(G)$. My Attempt: Clearly $y$ commutes with $t$, so $y$ commutes with $t^2$ as ...
### A group presentation given by the Collatz sequence of a fixed $n$. Has this been studied before?
This is a reference-request question. The Set Up: Fix $n\in\Bbb N$. Suppose $$C(x)=\begin{cases} x/2&:x\text{ is even},\\ 3x+1&:x\text{ is odd}. \end{cases}$$ Let $m\in\Bbb N\cup\{0\}$. Denote ...