Questions tagged [infinite-groups]

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

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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 ...
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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: ...
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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 ...
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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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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 $\left(1,1,1,\ldots \right)$ ...
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 ...
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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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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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 ...
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 ...
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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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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 $\omega$-...
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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 ...
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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 ...
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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 $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
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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 ...
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 ...
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. ...