Questions tagged [infinite-groups]

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

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7
votes
1answer
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 ...
10
votes
4answers
2k views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that $PSL_n(...
4
votes
2answers
484 views

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ? ( I know that there 'is' a 'surjection' , but I don't know whether any surjective homomrophism from $\mathbb R$ ...
8
votes
2answers
610 views

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 ...
26
votes
3answers
4k views

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 ...
11
votes
6answers
1k views

Does every infinite group have a maximal subgroup?

$G$ is an infinite group. Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ? Is it possible that there exists such series $H_1 < H_2 < H_3 <\cdots &...
3
votes
2answers
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 ...
0
votes
2answers
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. ...
27
votes
1answer
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 ?
13
votes
1answer
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?
13
votes
1answer
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)?
14
votes
1answer
144 views

Is there some sort of classification of all minimal non-cyclic groups?

Does there exist some sort of classification of all minimal non-cyclic groups (non-cyclic groups, such that all their proper subgroups are cyclic) I know the following classes of such groups: 1) $...
11
votes
1answer
321 views

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 $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ ...
12
votes
3answers
339 views

Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.

I'm trying to prove that the following group has infinite order: $$H=\langle a,b\mid a^{3}=b^{3}=(ab)^{3}=1\rangle.$$ Currently I'm checking on some cases using the relations, but my problem is the ...
4
votes
2answers
698 views

Finitely presented Group with less relations than Generators.

I some how feel that, any finitely presented group with less relations than Generators has to be an infinite Group. In One of my Questions, In an answer I have seen a group which is finitely ...
4
votes
3answers
280 views

Does there exist any surjective group homomorphism from $(\mathbb R^* , .)$ onto $(\mathbb Q^* , .)$?

Does there exist any surjective group homomorphism from $(\mathbb R^* , .)$ (the multiplicative group of non-zero real numbers) onto $(\mathbb Q^* , .)$ (the multiplicative group of non-zero rational ...
3
votes
2answers
160 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
4
votes
1answer
991 views

Infinite abelian group counterexample [duplicate]

A finite group $G$ is abelian iff all its irreducible representation $\rho$ have dimension 1. I'm looking for a counter-example when $G$ is an infinite group. Are there any? EDIT We're dealing with ...
4
votes
1answer
138 views

For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?

For any integers $m,n$ , both greater than $1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
2
votes
1answer
182 views

Is this an isomorphism possible?

I am working on the following homework problem: Let $\phi$ be an isomorphism from $\mathbb{R}^*$ to $\mathbb{R}^*$ (nonzero reals under multiplication). Show that if $r>0$, then $\phi(r) > 0$. ...
1
vote
1answer
75 views

Torsion elements of a group aren't necessarily a subgroup [duplicate]

In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup. I knew that the set ...
14
votes
1answer
729 views

On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of ...
24
votes
9answers
5k views

Probability of selecting an even natural number from the set $\Bbb N$.

I confirmed on this thread that there are as many as even natural numbers as there are natural numbers. Question : Suppose I have selected a number $n \in \mathbb N$ , what is the probability that $...
9
votes
4answers
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, ...
12
votes
1answer
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 ...
7
votes
2answers
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 ...
2
votes
1answer
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 ...
6
votes
0answers
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 ...
6
votes
1answer
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 ...
6
votes
1answer
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.
5
votes
1answer
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 $|B|\geq 3$? ($AB=\{ab: a\in A, b\in B\}$)
3
votes
1answer
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 = \...
2
votes
1answer
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 ...
5
votes
2answers
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 ...
3
votes
1answer
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 ...
2
votes
1answer
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 $|H|=\alpha$)?
7
votes
4answers
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.?...
3
votes
1answer
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 ...
3
votes
0answers
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 ...
3
votes
2answers
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)....
2
votes
0answers
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 ...
2
votes
3answers
103 views

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 ...
2
votes
0answers
34 views

If $A$ is a free abelian normal subgroup, and $K'$ has a unipotent image then $C_A(K')$ is non-trivial

Here's the context of the question: Let $G$ be an infinite polycylic group. It is known that there is $A \triangleleft G$ with $A \cong \mathbb{Z}^d$ for $d >0$. Define a homomorphism $\phi: G \...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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 $G$, ...
1
vote
0answers
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 \...
1
vote
3answers
63 views

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 $x\in X$ ). ...
-1
votes
1answer
44 views

If an infinite group acts freely on two sets 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 $x\in X$ ). ...