Articles of group theory

Abelian $2$-groups

Is every abelian group $A$ where every element has order two isomorphic to a direct product of cyclic groups of order two, $A\cong C_2\times C_2\times\ldots$? I ask because I used this “fact” in one of my old answers here (which is relevant to some work I am doing), and have just realised that this is […]

Infinite Groups with Finitely many Conjugacy Classes

If $G$ is an infinite group with finitely many conjugacy classes, what can be said about $G$? (Should $G$ be simple/ solvable/….?) For $n\geq 2$, does there exists a (infinite) group $G$ with exactly $n$ conjugacy classes, which is periodic also? I couldn’t find any information on these questions. One may provide links also for […]

Isomorphic Free Groups and the Axiom of Choice

When I read about free group, the proof which concerns about two free groups $F(X)$ and $F(Y)$ are isomorphic only if $\operatorname{card}(X) = \operatorname{card}(Y)$ has a sentence going as follows: $|M(X \cup X^{-1})|=|X \cup X^{-1}|=|X|$, using the axiom of choice. Can someone give me more hint about this question or some references?

Non trivial Automorphism

This question already has an answer here: $|G|>2$ implies $G$ has non trivial automorphism 2 answers

Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center

A nonabelian group $G$ of order $pq$ where $p$ and $q$ are primes has a trivial center My Proof is as follows: Assume we have nonabelian group $G$ of order $pq$ where both $p$ and $q$ are primes. When $G$ has a trivial center it means subgroup $Z(G)=\{e\}$. If a group is of order $pq$ […]

Permutation isomorphic subgroups of $S_n$ are conjugate

Consider $G,H \leq S_n$ and their natural action on $[n] = \{1,\ldots,n\}.$ We say that $G$ and $H$ are permutation isomorphic if there is a bijection $\varphi:[n] \mapsto [n]$ and group isomorphism $f:G \mapsto H$ so that $$\varphi(g(o)) = f(g) (\varphi(o))$$ or in the standard notation involving group actions $\varphi(o^g) = \varphi(o)^{f(g)}.$ I would like […]

why is a polycyclic group that is residually finite p-group nilpotent?

I am trying to solve an exercise in D. Robinson’s book A Course in the Theory of Groups, which asks me to show that if $G$ is polycyclic and residually finite p-group for infinitely many prime p, then $G$ is nilpotent and finitely generated torsion-free. How do I show the nilpotent part? The hint given […]

Any periodic abelian group is the direct sum of its maximal p-subgroups

I have an exercise. I cannot solve. Please help me to solve it: Prove that any periodic abelian group is the direct sum of its maximal $p$-subgroups.

Proof of $G=N\rtimes H$ iff $G=NH$ and $N\cap H=\{1\}$

I have the following definition of a semi-direct product: Let $G$ be a group. Suppose $N\triangleleft G$ and $H<G$ such that every element of $G$ can be uniquely written $g=nh$. Then $G$ is the semi-direct product of $N$ and $H$. I have to prove the following lemma: Let $N \triangleleft G$ and $H<G$. Then $G=N\rtimes […]

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does the remaining generator map to under the representation? 2) Why does a representation extend if and only if $z^{-1} = z^{p^{d}}$ for some […]