# Questions tagged [infinite-groups]

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

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### Generalisation of integers for infinite length?

As most people know, an integer can only be finite length when expressed in the form of a series of digits in some base. However, real numbers in general can be infinite length, so long as there is a "...
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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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### countable group, uncountably many distinct subgroup?

I need to know whether the following statement is true or false? Every countable group $G$ has only countably many distinct subgroups. I have not gotten any counter example to disprove the statement ...
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### Is it possible that a left coset of $H$ contains more than one right coset of $H$?

Let $H$ be a subgroup of group $G$. Is it possible that a left coset of $H$ contains more than one right coset of $H$? It is clear to me that the answer is 'no' if we deal with finite groups.
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### Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers?

Let $G$ be an infinite subgroup of $(\mathbb C \setminus \{0\},.)$ such that $G$ is compact as a subset of $\mathbb C$ , then is it true that $G=S^1$ ? I know that $G \subseteq S^1$ ; and since any ...
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### 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?
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### Automorphisms of infinite abelian groups

It is well-known that the map $\operatorname{Aut}$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ ...
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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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### Infinite Sets and Natural Numbers

Is it possible to find two infinite sets of non-negative integers, $A,B$, such that every non-negative integer can be written uniquely as a sum of two integers, one from $A$ and the other from $B$? ...
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### Can $\mathbb{R}$ be written as an ascending union of proper additive subgroups?

Can the group $\mathbb{R}$ be written as countable ascending union of proper subgroups? (i.e. does there exists a series of proper subgroups $H_1\leq H_2\leq \cdots$ such that $\cup {H_i}=\mathbb{R}$?...
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### Homogeneous groups

Let's call a group $G$ homogeneous if for every two distinct, non-identity elements $a$ and $b$ there is an automorphism $\phi$ of $G$ such that $\phi(a)=b$. Examining this definition, we can see ...
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### Is $\mathbb{Z}_2 \times \mathbb{Z}_4^\infty$ isomorphic to $\mathbb{Z}_4^\infty$?

Is $\mathbb{Z}_2 \times \mathbb{Z}_4^\infty$ isomorphic to $\mathbb{Z}_4^\infty$? My intuition says no, but I have not been able to find a proof. It is true that each group can be embedded in the ...
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### If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$? [duplicate]

If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$ ? This means, $gHg^{-1}$ is a proper subgroup of $H$. We know that $H \cong gHg^{-1}$, so if $H$ is ...
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### How do the subgroups of $(\mathbb C,+)$ which are connected (path connected) in $\mathbb C$ look like?

How do the subgroups of $(\mathbb C,+)$ which are connected in $\mathbb C$ look like ? Can they be characterized in some way ? (like for example , subgroups of $(\mathbb C\setminus \{0\} , .)$ , that ...
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### For general $n \in \Bbb N$ , how to determine all groups (both finite and infinite) having exactly $n$ conjugacy classes?

I was trying to solve the following problem : Determine all the finite groups having exactly 3 conjugacy classes. My attempt: Among all abelian groups $(\Bbb Z_3,+)$ only satisfies the property , ...
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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 ...
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### Is every infinite set equipotent to a field? [duplicate]

For example, $\mathbb N$ is equipotent to $\mathbb Q$ which is a field. $\mathbb R$ is equipotent to itself, which is a field. But what about $\mathbb R^{\mathbb R}$, $P(\mathbb R^{\mathbb R})$ etc.?...
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### Group Isomorphic to $(\mathbb{Z} \times \mathbb{Z})/ \langle(a,b),(c,d)\rangle$

I have come across an interesting question regarding quotient groups of $\mathbb{Z} \times \mathbb{Z}$: Suppose $(a,b)$ and $(c,d)$ are two independent elements of $\mathbb{Z} \times \mathbb{Z}$. ...
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### How can group theory explain movement on a hexagonal tiling?

(As a prelude, I have no formal math training other than high school. I am a beginner with group theory and have just recently begun picking it up and seeing its potential uses.) Imagine an infinite ...
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### Groups with 3 conjugacy classes and finite exponent

I have seen the question on groups with two conjugacy classes, and I proved to myself that such a group must be torsion-free (if it isn't the cyclic group of order 2), but what about a group with ...