# Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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### When is a quotient group $G/H$ abelian?

So clearly if $G$ is abelian and $H$ is a normal subgroup of $G$ then $G/H$ is abelian since $$xH.yH =(xy)H=(yx)H=yH.xH$$ But is there cases when this quotient group is abelian without the group G ...
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### Center of the dihedral group with odd and even number of vertices

I have posted a proof below, and would appreciate it if someone could review it for accuracy. Thanks! Problem: Let n $\in$ $\mathbb{Z}$ with $n$ $\ge$ 3. Prove the following: (a) Z(D$_{2n}$) = 1 ...
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### Prove that if $H,K \leq G$ where $G$ hyperbolic, $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate

Let $G = \langle S \mid R \rangle$ be a finite presentation, $G$ is $\delta$-hyperbolic. Prove that if $H,K \leq G$ where $H,K \cong C_2 \times C_2$, we can decide if $H$ and $K$ are conjugate. I am ...
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### $\mathbb{Z}^2 \ast \mathbb{Z}^2$ is isomorphic to no finite index proper subgroup of itself

I want to prove that $G = \mathbb{Z}^2 \ast \mathbb{Z}^2$ is isomorphic to no finite index proper subgroup of itself Here is my partial attempt: Consider the Bass-Serre tree $T$ that $G$ acts on in ...
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### Why is the symmetry of an equilateral triangle D3 and not D3h?

On this wiki page, it is explained that the symmetry of an equilateral triangle is $D_3$; the equilateral triangle is symmetric under the operations of $D_3$. The operations for $D_{3h}$ symmetry are ...
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### Prove that all proper subgroups of a group of order $8$ are commutative.

Prove that all proper subgroups of a group of order $8$ are commutative. Let $H$ be a subgroup of a group $G$. Then $o(H)\mid o(G)\implies o(H)=1,2,4$. If $o(H)=1$ or $2$, then $H$ is commutative. ...
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### A theorem in the paper “Noncommuting Random Products” by Furstenberg

I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963. The statement is as follows: Let $\mu$ ...
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### Suppose that $G$ is non-abelian group and $|G|=pq$ such that $p$ and $q$ are prime numbers [on hold]

Suppose that $G$ is a non-abelian group and $|G|=pq$ such that $p$ and $q$ are prime numbers and $N$ is normal subgroup of $G$ such that $|N|=q$ . show that $G'=N$.
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### GAP code to calculate the a certain subgroup $E(G)$ of a group

I am a research scholar from India. At present, I am working on a problem. For this problem, I need to construct the subgroup $E(G)$ of a group $G$ in GAP. Please help me. My question is as follows: ...
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### Is this a well-known group? $\langle a,b \mid a^5=b^4=e,b^{-1}ab=a^{-1}\rangle$

Consider the group $$G=\langle a,b \mid a^5=b^4=e,b^{-1}ab=a^{-1}\rangle$$ It looks like a dihedral group but it is not isomorphic to a dihedral group. Is this a well-known group?
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### Choice of symbols: $O_p(G)$, $O^p(G)$, and $O_\infty(G)$

For a finite group $G$ and a prime number $p$, several normal subgroups are defined as follows: $O_p(G)$ = the largest normal $p$-subgroup of $G$ ($p$-core) $O^p(G)$ = the smallest normal subgroup $N$...
I'm trying to construct a character table for a group of order 54 given by: $$\langle a,b : a^9 = b^6 = 1, b^{-1} a b = a^2\rangle$$ To do this first I need to compute conjugacy classes. This ...