# Tagged Questions

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 studies groups.

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### An example of a residually finite group which is not Hopf

trying to think of any residually finite group which is not Hopf. Any help?
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### How is a group made up of simple groups?

I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers. I've recently started ...
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### How can you show there are only 2 nonabelian groups of order 8?

It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. I've never ...
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### $A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
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### Theorems with the greatest impact on group theory as a whole

In his Contemporary Abstract Algebra text, Gallian asserts that Sylow's Theorem(s) and Lagrange's Theorem are the two most important results in finite group theory. He also provides this quote by G.A....
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### If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.

Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. ...
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### Why the term and the concept of quotient group?

The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as ...
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### A special subgroup of groups of order $n$

Let $G$ be a group with $|G| = n$ and let $\emptyset \ne S \subseteq G$. I want to show that $S^n$ is a SUBGROUP of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$.
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### Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ ...
### Finite number of subgroups $\Rightarrow$ finite group
I'm trying to prove that any group $G$ of infinite order has an infinite number of subgroups. I think that if the group has an element of infinite order, then it's easy because I can take the groups ...
### Subgroup of $\mathbb{Q}$ with finite index
Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$. I just saw this on our exam earlier and was stumped ...