Questions tagged [infinite-groups]

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

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1answer
46 views

Showing that finding all subgroups of an (infinite) group is undecidable.

Motivation: In trying to answer this question, I learnt that the question of determining the subgroups of finite groups is decidable. I wrote a short argument as a (now deleted) answer: Suppose ...
1
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0answers
37 views

Question about a group-theoretic Chinese remainder theorem and intersections of normal finite-index subgroups

I have recently been wondering about the group-theoretic Chinese remainder theorem. In particular, I was wondering whether this analogue can be generated to the case of more than two subgroups: ...
4
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3answers
82 views

Isomorphic subgroups such that the quotient is isomorphic.

Does there exists an infinite group $G$ which admits a normal subgroup $H$ such that $G,H,G/H$ are all isomorphic groups? If I don't remember badly, some time ago I read that the additive group of $\...
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0answers
19 views

Can all nonlinear groups be represented by integral transforms?

This is an extension to the infinite-dimensional case of the usual question, "are all groups linear". Given that nonlinear groups exist, can such groups always be represented as a group of integral ...
2
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0answers
24 views

Do the finite groups of Lie type have infinite analogs?

Of the 18 infinite families of finite simple groups, I know that $\mathbb{Z}$, the infinite analog of $\mathbb{Z}_p$, isn't simple while $A_\infty$, the one for $A_n$, is. What about the 16 families ...
4
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1answer
40 views

Random walk on infinite group - Entropy Speed and Growth

Let $\Gamma$ be an infinite, finitely generated group, and $X_n$ a random walk over $\Gamma$ with step distribution $\mu$. recall the definitions of : Entropy of $X_n$ : $\frac{-\log \mu^{*n}(X_n)}{...
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1answer
63 views

Infinite group whose every element is of order $4$?

I have constructed infinite group whose every element is of prime order by taking the set as set of sequences whose elements are from integers modulo $p$ and operation is integers modulo $p$. Now ...
8
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3answers
144 views

Can the group $\mathbb Z \times \mathbb Z$ be written as union of finitely many proper subgroups of it?

I want to know if the group $G=\mathbb Z \times \mathbb Z$ can be written as union of finitely many proper subgroups of it ? It is clear that $\mathbb Z$ can't be written as union of finitely many ...
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1answer
31 views

Does every element of $ \Pi_{n=1}^\infty{\mathbb{Z}_i} $ have finite order?

Does every element of $ \Pi_{n=1}^\infty{\mathbb{Z}_i} $ have finite order? If I took an infinite direct product of $\mathbb{Z}_i$ for $i=1,2,3,\ldots.$ Would the element $\left(1,1,1,\ldots \right)$ ...
8
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1answer
74 views

Infinite group with finitely many conjugacy classes of cardinality $n$.

Does there exist an infinite group $G$ such that: There are no conjugacy classes containing infinitely many elements. For every $n \in \mathbb{N}$, there are only finitely many conjugacy classes ...
2
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1answer
36 views

Does there exist a measure on an infinite cyclic group that assigns each coset its natural density?

Suppose, $F$ is the minimal $\sigma$-algebra on an infinite cyclic group $C_\infty = \langle a\rangle_\infty$, that contains all cosets of non-trivial subgroups of $C_\infty$. Does there exist such a ...
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1answer
70 views

The tree of $\langle a,b|a^2=b^3\rangle$.

I want to draw the usual tree that $G=\langle a,b|a^2=b^3\rangle$ acts on. EDIT THIS IS WRONG So I wrote $G=\mathbb{Z}_2*\mathbb{Z}_3=\langle a\rangle*\langle b\rangle$. I set $G_0=\{e\}$ the ...
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0answers
35 views

Properties of the set of all reducible polynomials?

Consider the set of all reducible integer-valued polynomials in $\mathbb Z$, meaning all those which can be factored into at least two integer-valued parts. My question is as to the sort of properties ...
3
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1answer
51 views

Subgroups of infinite $p$-group ($p$ is prime)

I am doing exercise which is a problem from the book Algebra by Hungerford (exercise II.5.4). Given an infinite $p$-group $G$, where $p$ is a prime, I need to show that either $G$ has a subgroup of ...
3
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1answer
51 views

Are infinite symmetric groups always equal to their word maps?

Suppose $G$ is a group $w \in F_\infty$, where $F_\infty$ is the free group of countable rank. Let’s define the corresponding word map as $w(G) := \{g \in G| \exists f \in Hom(F_\infty, G) f(w) = g\}$,...
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1answer
43 views

Is the class of $\omega$-soluble groups a variety?

Let’s call a group $G$ $\omega$-soluble if $\bigcap_{i = 1}^{\infty} G^{(i)}$ is trivial. Here $\{G^{(i)}\}_{i = 1}^\infty$ stands for the derived series of the group. Note, that not all $\omega$-...
2
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1answer
52 views

Isomorphism between two finitley presented groups which are not finite

I am looking for an isomorphism between the two following groups (infinite groups) ( probably by GAP to show that there is an isomorphism between $f$ and $g$) $$f=\langle a,b,c ~| ~ bab^{-1}a^{-1},...
13
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1answer
181 views

Can a group have a cyclical derived series?

Given any group $G$, one can consider its derived series $$G = G^{(0)}\rhd G^{(1)}\rhd G^{(2)}\rhd\dots$$ where $G^{(k)}$ is the commutator subgroup of $G^{(k-1)}$. A group is perfect if $G=G^{(1)}$ ...
1
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1answer
103 views

why the multiplication between functionals is not well defined?

I'm reading Olver (2012), chapter 7, and it says that the multiplication between functionals is not well defined. Then he refers to the section 5.4 where gives an explanation why, but I can't ...
2
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1answer
55 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, ...
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0answers
84 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 $...
2
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0answers
52 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 ...
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0answers
33 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
54 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 ...
2
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0answers
58 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
94 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
77 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 $(\...
14
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1answer
151 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
50 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 ...
0
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3answers
40 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, ...
5
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0answers
54 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 ...
2
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0answers
34 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
42 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
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1answer
36 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 ...
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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 ...
6
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2answers
80 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
52 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
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1answer
61 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, .. ...
7
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0answers
200 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 $...
16
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4answers
163 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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1answer
61 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 ...
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0answers
123 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 $...
0
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1answer
67 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\...
2
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1answer
55 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
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1answer
79 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
32 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
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1answer
22 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
63 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 ...
15
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2answers
895 views

Can an uncountable group have a countable number of subgroups?

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 ...
0
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2answers
104 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. ...