Articles of infinite groups

Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$

Define $H = \{(a, 0): a\in \mathbb{R}\}$. Without using the fundamental homomorphism theorem, how would we know what $\mathbb{R}^2/H$ is? The quotient group is $\{H, (x, y) + H, (x_2, y_2) + H, \dots \}$. Intuitively, each coset in the quotient group is a horizontal line crossing through the point $(x_i + a, y_i)$ or […]

How to classify all the abelian groups with finite exponent?

Let $A$ be an abelian group, the exponent exp$A$ is the least natural number $n$ (if exists) such that $nA=0$ or $+\infty$. The question can be reduced to the case exp$A=p^n$ for a certain prime number $p$ easily. But then I have no ideal. I’m not very sure if such a group must be a […]

Is the center of a p-group non-trivial?

Is it true that if $G$ is a $p$-group, where $p$ is a prime, then center of $G$ is non-trivial? I know for finite $p$-group center of $G$ is non-trivial (easy to prove using class equation for group). But I am not sure about infinite $p$-group. I have no idea how to approach this problem. […]

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 […]

Is every infinite set equipotent to a field?

This question already has an answer here: Fields of arbitrary cardinality 5 answers

Subgroups of an infinite group with a given index

Let $G$ be an infinite group and $\alpha$ a cardinal number with $\aleph_0\leq \alpha\leq |G|$. Is there a subgroup $H$ of $G$ with $|G:H|=\alpha$ (what about $|H|=\alpha$)?

Is there any non-abelian group with the property $AB=BA$?

Is there any finite (resp. infinite) non-abelian group of order $\geq 8$ such that $AB=BA$ for all subsets $A, B$ with $|A|\geq 3$ and $|B|\geq 3$? ($AB=\{ab: a\in A, b\in B\}$)

An infinite $p$-group may not be nilpotent

It is well-known fact that every finite $p$-group $G$ is nilpotent. I am asking to have a counter example when $G$ is infinite $p$-group. Thanks.

Groups with finite automorphism groups.

An easy argument shows that for any finite group $G$ the cardinal of $Aut(G)$ is less than $(|G|-1)!$. In particular the automorphisms group of a finite group is finite. Basically my question is about the converse statement. If a group $G$ has finite automorphisms group then should $G$ be finite ? The answer to this […]

Torsion Subgroup (Just a set) for an abelian (non abelian) group.

Let $G$ be an abelian Group. Question is to prove that $T(G)=\{g\in G : |g|<\infty \}$ is a subgroup of G. I tried in following way: let $g_1,g_2\in T(G)$ say, $|g_1|=n_1$ and $|g_2|=n_2$; Now, $(g_1g_2)^{n_1n_2}=g_1^{n_1n_2}g_2^{n_1n_2}$ [This is because G is abelian]. $(g_1g_2)^{n_1n_2}=g_1^{n_1n_2}g_2^{n_1n_2}=(g_1^{n_1})^{n_2}(g_2^{n_2})^{n_1}=e^{n_2}e^{n_1}=e$ Thus, if $g_1,g_2$ have finite order, so is $g_1g_2$.So, $T(G)$ is closed under […]