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Questions tagged [infinite-groups]

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

0
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3answers
51 views

How to prove that rational numbers are countable to someone who won't accept the visual proof? [closed]

I am trying to prove to someone that the rational numbers are countable by using the proof shown on the website linked here: https://www.homeschoolmath.net/teaching/rational-numbers-countable.php. ...
2
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0answers
46 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
84 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
votes
2answers
63 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 $(\...
13
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1answer
130 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
45 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
33 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
52 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
30 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
1answer
38 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
33 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
42 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
votes
2answers
66 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
vote
1answer
43 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
32 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, .. ...
6
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0answers
154 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 $...
15
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4answers
148 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
vote
1answer
56 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 ...
1
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0answers
118 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
44 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
49 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
67 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
25 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
votes
1answer
48 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 ...
14
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2answers
840 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 ...
0
votes
2answers
78 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. ...
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
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1answer
40 views

If there exists an epimorphism from a group $G_1$ to $G_2$ that is not one-to-one, then can $G_1$ and $G_2$ be isomorphic? [duplicate]

I am trying to show that two groups are isomorphic only if a certain condition holds. I can show that a specific epimorphism between the two groups is one-to-one only if this condition holds, but it ...
0
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0answers
80 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
2
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1answer
27 views

Definitions of normalisers for infinite groups

If G is a group and A is a subset of G, the normaliser of A in G can be defined as either (1) $N_G(A) = \{g \in G\ |\ gag^{-1} \in A, \forall a \in A\}$ (2) $N_G(A) = \{g \in G\ |\ gAg^{-1} = A \}$ ...
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0answers
31 views

Are there more real numbers than irrational numbers? [duplicate]

Wrapping my head around the mathematical definition of infinity and just curious here: Are there more real numbers than irrational numbers? It would intuitively seem so, but they are both just ...
0
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1answer
27 views

An infinite left word in some alphabet

What is the infinite left word in some alphabet? I can understand the definition of the infinite right word -- some path in Cayley graph from $id$-element to $+\infty$ or $-\infty$. But is the ...
-1
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1answer
107 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
1answer
52 views

finding the geometric dimension of $\mathbb{Z}^n$

I've just been introduced to the idea of geometric dimension of a group (smallest dimension of $K(G,1)$) and I'm trying to figure out what the geometric dimension of $\mathbb{Z}^n$ is for $n\geq 3$. ...
0
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0answers
24 views

Confusion about infinite order groups, subgroups and comparing order between them.

I was given feedback for a homework problem, Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $a \in G$. Let $Ha$ be a subgroup of $G$. The problem was to show that $Ha = H$. I established ...
1
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1answer
26 views

Is there an infinite group generated by the elements of two (or more, but finitely many) normal subgroups?

It's perfectly easy to get an infinite group generated by the elements of a finite set of finite subgroups: take a free product. But is it possible to have an infinite group be generated by the ...
1
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1answer
61 views

Unitary matrices and Spectral Theorem

Due to the Spectral theorem and Shur's decomposition, if $A$ is a unitary matrix, then $$A = QDQ^{-1} \quad (1)$$ where $D$ is diagonal and $Q$ unitary. Now, let $A$ belongs to the center of SU(n) ...
1
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1answer
66 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 ...
0
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0answers
40 views

A subgroup of abelian torsion group with infinite exponent

Take an abelian torsion group, with infinite exponent, say, G=C2+C4+C8+..+C2^k+.., where Cn is cyclic of order n. What is the subgroup S that is the intersection of all subgroups of G with infinite ...
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0answers
59 views

Classifying elements of finite and infinite order in $GL_{2}(\mathbb{R})$

A problem on a recent assignment defined the Torsion subset $F(G)$ of a Group $G$ as the set of elements of G of finite order. It then asked to prove that $F(GL_2(\mathbb{R}))$ is not a subgroup of $...
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0answers
47 views

Infinite sum $S(k) = \lambda + 2^k \lambda^2 + 3^k \lambda^3 + \dots$

I'm trying to find an expression for: $$S(k) = \sum_{n=0}^{\infty} n^k \lambda^n$$ I have found a recursive expression for this where I first find $S(0)$, then use that to find $S(1)$. Then use those ...
3
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1answer
54 views

A question about the symbol $n$ in the additive group $\mathbb{R}^n$

As we know the symbol $n$ determines the dimension of $\mathbb{R}^n$ in the category of vector spaces. Also, in the category of manifolds, $n$ determines the dimension of $n$-manifold $\mathbb{R}^n$. ...
1
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1answer
56 views

About infinite chains of subgroups containing an infinite subgroup of a locally finite group

Let $G$ be an infinite locally finite (non-solvable) group, let $\{H_i\}_{i\in\mathbb{N}}$ be a strictly totally ordered family of subgroups of $G$ and let $H$ be the intersection of that family. If $...
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votes
2answers
44 views

About the set of periods of a group subset [closed]

Let $G$ be a group and $A\subseteq G$. Put $$ S(A):=\{g\in G: gA=A\}. $$ It can be shown that $S(A)$ is a subgroup of $G$, and $S(A)=A \ \iff \ A\leq G$. Now, is it true that: (1) $|S(A)|\leq |...
0
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1answer
98 views

Proof: $\kappa \cdot \kappa = \kappa$ for infinite cardinals [duplicate]

Im looking for a detailed proof for $\kappa \cdot \kappa = \kappa $ with $\kappa $ beeing a infinite cardinal number. The problem is described in a book (Frank R. Drake, Set Theory: An Introduction to ...
4
votes
1answer
59 views

Nilpotent groups and 2-generated subgroups

Do you know an example of a $2$-locally nipotent group $G$ which is not locally nilpotent? $2$-locally nilpotent: every subgroup which is generated by $2$ elements is nilpotent. locally nilpotent: ...
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0answers
26 views

Can the triangularization of a linear group preserve a given good feature of an element?

Let $k$ be an algebraic-closed field and let $G$ be a soluble subgroup of $GL_n(k)$. By a result of Mal'cev (Bertram A.F. Wehrfritz - Infinite Linear Groups, Theorem 3.6), $G$ contains a normal ...
2
votes
1answer
143 views

Are infinite groups permutation groups?

I have known Cayley's Theorem for some time now, which shows that all finite groups are permutation groups (secretly, as a previous mathematics teacher of mine might have put it). However, the thought ...
4
votes
0answers
40 views

Conjugacy class with $1$ element in an Infinite Simple Group

We would like to show that if $G$ is an infinite simple group, then the only conjugacy class of exactly one element is $\{1_G\}$. My thoughts: We want to proove that if $|\mathrm{orb}(x)|=1\iff \...