# 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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### If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is cyclic, ...
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### Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
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### Conjugate subgroup strictly contained in the initial subgroup?

Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$? If $H$ has finite index or finite order, this is ...
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### Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In ...
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### Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
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### The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a ﬁeld and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
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### Nonabelian semidirect products of order $pq$?

I just constructed the semidirect product in Lang, and I'm trying to tie some facts together. From Ash's Algebra, I know that if $p\lt q$ are distinct primes, if $q\not\equiv 1\pmod{p}$, then any ...
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### A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$

I would be very thankful if someone could give me a hint with proving this. It's a very common exercise in abstract algebra textbooks. If $G$ is a group with a subgroup $H$ of finite index $n$, ...
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### Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1$, then how do I prove $G$ is cyclic without using Sylow's theorems?
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### Structure of groups of order $pq$, where $p,q$ are distinct primes. [duplicate]

I don't know about the Sylow Theorems. But I have been wondering about a proof of the fact that a group or order $pq$ where $p$ and $q$ are distinct primes must be cyclic. I can't quite work out the ...