Questions tagged [infinite-groups]

For questions about groups where the underlying set has infinite cardinality.

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What is $\operatorname{Aut}(\mathbb{R},+)$?

I was solving some exercises about automorphisms. I was able to show that $\operatorname{Aut}(\mathbb{Q},+)$ is isomorphic to $\mathbb{Q}^{\times}$. The isomorphism is given by $\Psi(f)=f(1)$, but ...
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Example of an infinite group where every element except identity has order 2

Find an infinite group, in which every element g not equal identity (e) has order 2 Does this question mean this: the group that fail condition (2) which is no inverse and also that group must have ...
212 views

Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.

Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group. A hints is given as order of G is not infinite since $a$ and $a^{-1}$ are only generators. ...
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Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?
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Does $A^{-1}A=G$ imply that $AA^{-1}=G$? [closed]

Let $G$ be a group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $A^{-1}A=G$ then $AA^{-1}=G$ (and vice versa)?
434 views

Countable number of subgroups $\implies$ countable group

I know that if a group $G$ has a finite number of subgroups then the group $G$ is finite. But if a group $G$ has countable number of subgroups then is the group countable?
659 views

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 ...
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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, ...
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Finitely generated group has only finitely many subgroups of given index

In my previous question i have got a comment that : If a group is finitely generated, then there are finitely many subgroups of a given finite index. I do not yet see the beauty of this problem ...
454 views

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.
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Examples of Groups (resp. Rings, Fields, etc.) Which Are Isomorphic to a Proper Subgroup (resp. Subring, Subfield, etc.)

Was just reading this question When is a group isomorphic to a proper subgroup of itself? and was wondering not about the conditions for being isomorphic to a proper subgroup, subring, etc., but about ...
224 views

$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?

Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ? Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in ...
199 views

Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? [duplicate]
Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? It is rater obvious, that if such group exists then it is generated by any ...