Questions tagged [infinite-groups]

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

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Infinite orders (group theory) as cardinals

Let be $(G,\cdot)$ an infinite group. Usually, the order of an element $a\in G$ defined as the smallest positive integer such that $a^n=1.$ Background: The equality $[G:K]=[G:H][H:K]$ for $K\le H\...
26
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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 ...
24
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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 $...
2
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1answer
52 views

Infinite $\frac{3}{2}$-generated groups?

Let’s call a group $\frac{3}{2}$-generated iff for every $g \in G \setminus \{1\}$ there is some $h \in G \setminus \langle g \rangle$ with $G = \langle g,h \rangle$. There is a conjecture by Breuer, ...
1
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0answers
81 views

Does $\Sigma$ generate the variety of all groups?

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
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0answers
199 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 $...
2
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0answers
50 views

Does there exist a nontrivial abelian group $A$ such that $\mathrm{Aut}(A) \cong A$? [duplicate]

Does there exist a nontrivial abelian group $A$, such that $\mathrm{Aut}(A) \cong A$? Here $\mathrm{Aut}$ stands for the automorphism group. What do I currently know: If such $A$ exists, it has ...
11
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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 $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ ...
0
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0answers
29 views

Is the unitary group $U(2) \cong SU(2) \times T$ [duplicate]

Let $U(2)$ denote the group of all invertible $2×2$ complex matrices $A$ with $A \overline{A}^T=I$ where $T$ denotes transpose matrix. Let $SU(2)$ be the subgroup of $U(2)$ consisting of those ...
2
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0answers
50 views

Abelian groups having no finite subgroup

My question is: Are there currently any classification (up to isomorphism) of infinite abelian groups having no non-trivial finite subgroup? It seems many have asked whether an infinite group can ...
14
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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) $...
2
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0answers
52 views

How to enumerate Von Dyck groups?

I would appreciate an algorithm to list all elements of a given Von Dyck group $D(p,q,r)$, each once, in a format that would allow me to find compositions and inversions within that list. ...
1
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2answers
87 views

Commensurability classes in groups

Suppose $G$ is a group. Let’s call $a, b \in G$ commensurable, iff $\exists m, n \in \mathbb{Z} \setminus \{0\}$, such that $a^n = b^m$. One can see, that commensurability is an equivalence ...
2
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2answers
70 views

Why not $\mathbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}\cong \mathbb{Q}^2$ not isomorphic as additive groups?

We know that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups(as they are isomorphic as $\mathbb{Q}$ vector spaces) under the axiom of choice. But why not $(\mathbb{Q}, +)$ and $(\...
2
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2answers
301 views

Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$. For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of ...
2
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1answer
60 views

Index of subgroups in infinite p-group

If $G$ is a finite $p$-group, it is trivial that every subgroup has index $p^r$ for some integer $r$. If $G$ is infinite, this is not true as the index can be infinite. If $G$ is an infinite $p$-...
2
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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 ...
5
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0answers
53 views

Does there exist an Artinian verbally simple group, which is not characteristically simple?

Does there exist an Artinian verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally ...
0
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3answers
34 views

Finitely generated, non abelian, infinite group

I was making a diagram of different types of groups; finite /infinite, cyclic / non-cyclic, finitely generated / inifinitely generated, but realized that I didn't have any examples og infinite groups, ...
1
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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 ...
2
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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
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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 ...
2
votes
1answer
34 views

Subgroup of non-simple group necessarily non-simple?

Let $G$ be an infinite group that is not simple. If $B$ is a subgroup of $G$, must $B$ necessarily be not simple as well? I know that if $N$ is some proper normal subgroup of $G$, then $B\cap N$ is a ...
6
votes
2answers
71 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 ...
1
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1answer
45 views

Let G be a group with composition series. Let $H\triangleleft G$. Then $H$ is in some of the series.

I know this is true for $G$ being finite. How about infinite group? Is this statement still valid? I proved the finite case by induction on $|G|$ (which supposedly not valid for infinite case) ...
4
votes
1answer
40 views

Is the free product of residually finite groups always residually finite?

Suppose groups $G$ and $H$ are residually finite. Does that imply, that $G \ast H$ is residually finite? What have I tried to prove this: Suppose, $a = g_1h_1g_2h_2…g_nh_n \in G \ast H$, $g_1, .. ...
16
votes
4answers
154 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 ...
1
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0answers
122 views

Torsion-free abelian groups of finite rank and a subgroup of finite index (Fuchs' problem) - self study

I'm trying to solve the following exercise (Fuchs, "Infinite Abelian Groups", Vol. $2$, pp. $153$, Ex.$5$): Let $A$ be a torsion-free abelian group of finite rank. If $\phi$ is an isomorphism of $...
4
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0answers
122 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
2
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2answers
381 views

Residually finite nilpotent group

It is known that every finitely generated nilpotent group is residually finite. Why finitely generated hypothesis is essential?
2
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1answer
216 views

Torsion-free nilpotent Group

I am looking for a short proof to the following fact: In torsion-free nilpotent group we have: an non-trivial element cannot be conjgate to it inverse. I know very little about nilpotent ...
3
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2answers
172 views

Would you please construct a locally nilpotent group which is not nilpotent?

Would you please construct a locally nilpotent group which is not nilpotent? An example of a locally nilpotent group which is not nilpotent?
5
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1answer
771 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 ...
1
vote
1answer
59 views

Are infinite indexes multiplicative?

For a group $G$ and its subgroup $H$, the index of $H$ relative to $G$, denoted by $[G:H]$, is the cardinality of the set $\{ gH \mid g \in G \}$. It is known that $|G| = [G:H]|H|$ even if all ...
0
votes
1answer
59 views

Direct sums and products of SU(2) representations

I am reading the book on group theory and stuck with a simple problem. Why $$(2\bigotimes2)\bigoplus(2\bigotimes1)\bigoplus(1\bigotimes2)\bigoplus(1\bigotimes1)=3\bigoplus1\bigoplus2\bigoplus2\...
3
votes
0answers
63 views

Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$

Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Lets say the generators of ...
2
votes
1answer
51 views

When group action on cartesian product is transitive?

Let $G$ be a group acting transitively on sets $\Omega$ and $\Lambda$. Then there is a natural induced action of $G$ on cartesian product $\Omega \times \Lambda$. I can prove that if $G$ is finite ...
5
votes
1answer
72 views

Let $G$ be a polycyclic group and assume that every finite quotient of $G$ is nilpotent. Then $G$ is nilpotent

Let $G$ be a polycyclic group and assume that every finite quotient of $G$ is nilpotent. Then $G$ is nilpotent. First some preliminaries: Every infinite polycyclic group contains a free ...
2
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0answers
26 views

Index of a common normal core

Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
0
votes
1answer
19 views

Countable Infinite and Uncountable Infinite sets

Mark each statement as TRUE, FALSE, or UNKNOWN (a) $|\Bbb{R}| < \aleph_1$ (b) $|\Bbb{R}| = \aleph_1$ (c) $|P(\Bbb{R})| > \aleph_1$ Could someone explain to me the reasoning based on whatever ...
0
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1answer
53 views

Commutator Subgroup of Thompson's Group $F$

Let $G$ be an infinite non-abelian group. Is there any statement like this in group theory : If its commutator subgroup is simple then $G$ is simple? Normally if $G$ is non-abelian, simple, its ...
3
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3answers
127 views

Show that $G/H\cong\mathbb{R}^*$.

Let $G:= \bigg\{\left( \begin{array}{ccc} a & b \\ 0 & a \\ \end{array} \right)\Bigg| a,b \in \mathbb{R},a\ne 0\bigg\}$. Let $H:= \bigg\{\left( \begin{array}{ccc} 1 & b \\ 0 & 1 \\...
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?
14
votes
2answers
854 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 ...
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 ...
0
votes
2answers
79 views

Show Circle Group $ \mathbb{T} $ isomorphic to $\mathbb { C } ^ { * } / \mathbb { R } ^ { * } $

I'm trying to think of an extension to this question, which asks to show whether $\mathbb { C } ^ { * } / \mathbb { R } ^ { + } \simeq \mathbb { T } $. They do it using the First Isomorphism Theorem. ...
4
votes
3answers
762 views

What group is $\mathbb{R}/\mathbb{Z}$ isomorphic to?

This is not so much a plea of ignorance, but rather me trying to see whether intuitively I actually understand what is going on in group theory. The question asks What group is $\mathbb{R}/\mathbb{Z}$...
2
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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 \...
-1
votes
1answer
132 views

$\omega$th iteration of Cayley-Dickson construction

The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -...
0
votes
0answers
81 views

Does $\aleph_0!=\omega$? [duplicate]

More generally, is the order type of some cardinal $\alpha$ equal to $\alpha!$? Related What is $\aleph_0!$? Factorial of Infinite Cardinal factorial of infinite Cardinals