# Questions tagged [infinite-groups]

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

50 questions
Filter by
Sorted by
Tagged with
927 views

### 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 ...
2k views

2k views

### 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 ...
204 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. ...
2k views

### 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 ?
417 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?
256 views

### 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)?
144 views

6k views

### 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, ...
632 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 ...
543 views

### 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 ...
204 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 ...
2k views

### 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 ...
216 views

### 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 ...
431 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.
179 views

### 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 $|B|\geq 3$? ($AB=\{ab: a\in A, b\in B\}$)
5k views

### SO(2) group generator Lie Algebra

For the $2 \times 2$ orthogonal group of matrices which for the $SO(2)$ group, there is only one free parameter in the group element and hence only one generator for the group. Which is,  X_g = \...
40 views

### Does there exist a verbally simple group, which is not characteristically simple?

Does there exist a verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple ...
1k views

### Probability that 3 points are collinear.

If we select three independent and random points $A$,$B$ and $C$ in a plane, what shall be the probability that they are collinear? Actually, this problem was asked to my friend in an interview, he ...
352 views

### Automorphisms of the Prüfer group

Let $p$ be a prime number. Can you give me a few examples of automorphisms of $\Bbb Z_{p^\infty}$ other than the identity function? I'm looking for an elemetary way to construct them. It can be ...
172 views

### 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 $|H|=\alpha$)?
516 views

### 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.?...
230 views

### 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 ...
234 views

### 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 ...
621 views

### 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)....
32 views

### Infinite Abelian Groups: Unnecessary Hypothesis in Kaplansky's Lemma 7

Looking for a validation of a proof of a stronger statement. The essential argument that needs review is the final paragraph. This would obviate the need for a much more complicated argument I gave ...
103 views

47 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 ...
45 views

### Correcting Proof of Lemma 10 in Kaplansky's *Infinite Abelian Groups*

Lemma 10 in Kaplansky's Infinite Abelian Groups states: Let $G$ be a primary group, $H$ a pure subgroup, $x$ an element of order $p$ not in $H$. Suppose that $h(x)=r<\infty$, and suppose ...
274 views

### Galois group of the field of all constructible complex numbers

I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call ...
206 views

### Is there a classification of f.g. abelian-by-finite groups? [EDITED]

Let $G$ be a f.g. abelian-by-finite group, i.e. there exists a f.g. abelian group $N$ which is normal in $G$ and such that the quotient $G/N = Q$ is finite. The problem of classifying all such $G$, ...
39 views

### Is conjugation in infinite groups well behaved? [duplicate]

A question from earlier today made me start to think of this question. Let $G$ be an infinite group with infinite subgroup $H$. Can the condition below ever hold? \begin{equation*} \exists g\in G \...
### If an infinite group $G$ acts freely on two sets of same cardinality $> |G|$, then the sets are bijective via an action preserving bijection?
Let $G$ be an infinite group. Let $X,Y$ be two sets on which $G$ acts freely. (An action of a group $G$ on a set $X$ is called free iff $G_x:=\{g\in G : gx=x\}$ is singleton for every $x\in X$ ). ...
Let $G$ be an infinite group. Let $X,Y$ be two sets on which $G$ acts freely. (An action of a group $G$ on a set $X$ is called free iff $G_x:=\{g\in G : gx=x\}$ is singleton for every $x\in X$ ). ...