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Questions tagged [infinite-groups]

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

29
votes
3answers
1k views

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 "...
26
votes
2answers
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 ...
26
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 ?
25
votes
2answers
699 views

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.
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 $...
15
votes
1answer
675 views

Is there a good example of a subgroup of an infinitely generated abelian group that is not isomorphic to a quotient of that group?

Whilst I understand the classification of the finitely generated abelian groups, this had me wondering whether there is a subgroup $H$ of a general (necessarily infinitely generated) abelian group $G$ ...
15
votes
4answers
149 views

Is there a group $G$ and subgroup $H$, such that there exists $g\in G$ with $gHg^{-1} \subset H$ and $|H:gHg^{-1}|$ is infinite?

Question (asking on behalf of my friend who studies abstract algebra): Is there a group $G$ and subgroup $H$, such that there exists $g\in G$ with $gHg^{-1} \subset H$ and $|H:gHg^{-1}|$ is ...
14
votes
2answers
850 views

Can an uncountable group have a countable number of subgroups? [closed]

Can an uncountable group have only a countable number of subgroups? Please give examples if any exist! Edit: I want a group having uncountable cardinality but having a countable number of ...
14
votes
1answer
725 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 ...
13
votes
3answers
611 views

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 ...
13
votes
1answer
412 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
163 views

Injecting a group into itself

Let $G$ be a group with elements $g_1, g_2\in G$ and injective endomorphisms $\phi_1, \phi_2$ s.t. $\phi_1 (g_1)=g_2$ and $\phi_2(g_2)=g_1$. Does this imply there is an automorphism $\psi$ with $\...
13
votes
1answer
255 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)?
13
votes
1answer
135 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) $...
12
votes
3answers
337 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 ...
12
votes
1answer
621 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 ...
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 &...
11
votes
1answer
314 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 $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ ...
11
votes
1answer
282 views

Is there a standard name for this infinite group?

Consider the group of sequences $$\{(a_1,a_2,\dots): a_i\in\mathbb{Z}/2\mathbb{Z}\}$$ where the group operation is component-wise addition. Is there a standard name for this group, such as $(\mathbb{Z}...
10
votes
3answers
275 views

Classify all groups containing isomorphic copy of $\mathbb{Z}$ of index $2$.

I have the following question: Classify all groups $G$ containing an isomorphic copy of $\mathbb{Z}$ such that the copy has index $2$ in $G$ There are some candidates: $\mathbb{Z}\times\mathbb{Z}/...
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(...
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, ...
9
votes
1answer
479 views

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 $B$? ...
9
votes
1answer
279 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups $...
9
votes
3answers
235 views

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 $\cup {H_i}=\mathbb{R}$?...
9
votes
2answers
791 views

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 ...
8
votes
1answer
229 views

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 ...
8
votes
2answers
600 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 ...
8
votes
1answer
235 views

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 ...
8
votes
0answers
75 views

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 , ...
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 ...
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.?...
7
votes
2answers
514 views

Weird isomorphisms of infinite groups

According to my interpretation to one of the answers in Splitting in Short exact sequence, $$\Bbb R \cong \Bbb Q \oplus \Bbb R / \Bbb Q$$ also, according to What is known about the quotient group $\...
7
votes
2answers
395 views

Automorphism group of the group $(\mathbb{R}^n,+)$

As a vector space, it is obvious and well-known that the automorphism group of $\mathbb{R}^n$ is $GL_n(\mathbb{R})$. My question is: What is the $\text{Aut}((\mathbb{R}^n,+))$ ($n \ge 2$) as a group? ...
7
votes
1answer
915 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 ...
7
votes
1answer
117 views

Are there non-abelian groups with the property $|AB|=|BA|$?

Regarding to the question "Is there any non-abelian group with the property $AB=BA$?", now it is important for us to know that: (a) Is there any finite (resp. infinite) non-abelian group of order $\...
7
votes
2answers
127 views

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 $\mathbb{Z} \times \mathbb{Z}$. ...
7
votes
0answers
189 views

Is $Z(\Sigma) \cong E$?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
7
votes
0answers
115 views

On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
6
votes
2answers
69 views

The number of elements not in conjugate of a subgroup

There is a proposition about the number of elements not lying in any conjugate of a subgroup when the group is finite. It states: If $G$ is a finite group and $H$ is a proper subgroup, then the ...
6
votes
1answer
429 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.
6
votes
1answer
236 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
6
votes
1answer
174 views

Subgroups of an infinite abelian group with a given index

This is followed by the question: Subgroups of an infinite group with a given index (with a counterexample of non-abelian groups). Now, the question is: Let $G$ be an infinite abelian group and $\...
6
votes
1answer
79 views

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 ...
6
votes
1answer
213 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
0answers
145 views

$H$ is a normal subgroup (containing an element of infinite order) of the permutation group over positive integers , then is $H$ the whole group?

Let $S(\mathbb N)$ be the set of all bijections on $\mathbb N$ ( the set of positive integers ) endowed with the natural group structure of function composition . If $H$ is a normal subgroup of $S(\...
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 ...
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 $|B|\geq 3$? ($AB=\{ab: a\in A, b\in B\}$)
5
votes
1answer
766 views

Nonabelian infinite nilpotent groups

Can you give examples of nonabelian infinite nilpotent groups? Here's what I got so far: The Heisenberg group. The free nilpotent group of class $s$ (thanks Arturo for your comment here). The group ...
5
votes
1answer
2k views

The Perfect Sharing Algorithm (ABBABAAB…)

Less of a question and more of an exercise, it has to do with something I found while doing some programming and being unable to find things. Basically I wanted a formula for the perfect sharing ...