# Questions tagged [infinite-groups]

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

66 questions with no upvoted or accepted answers
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### 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 , ...
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### 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 ...
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### 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 ...
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### 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 $F$ ...
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### 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 ...
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### $\mathbb{Z}$-linear reidually solvable groups

Is there a known important class of $\mathbb{Z}$-linear reidually solvable groups, except for free groups? It is just a rough question and I'm not an expert on this subject. I read some books on ...
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### The group of pure/hereditary sets

From wikipedia: In set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of ...
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### Infinite Groups with Finitely many Conjugacy Classes

If $G$ is an infinite group with finitely many conjugacy classes, what can be said about $G$? (Should $G$ be simple/ solvable/....?) For $n\geq 2$, does there exists a (infinite) group $G$ with ...
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### Properties of Finite and Infinite $p$-Groups

By a $p$-group, we mean a group in which every element has order a power of $p$. It is well known that finite $p$-group has non-trivial center. But, an infinite $p$-group may have trivial center. ...
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### Finitely generated infinite $p$-group with a unique subgroup of order $p$

I'd like to ask can we characterize the structure of finitely generated infinite $p$-group which has a unique subgroup of order $p$? Can we say that these group are residually nilpotent? Any ...
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### 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 ...
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### 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. ...
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### 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 ...
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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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### 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 ...
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### example of p-group with infinite Frattini subgroup

Is there an infinite $p$-group $G$ with infinite Frattini subgroup $\Phi(G)\not = G$? In "Subgroups of Teichmuller Modular Groups" there is an example but I don't get it because I don't know much ...
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### Galois group of the field of all constructible complex numbers

I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call ...
How to prove a locally cyclic group $(G,.)$ is isomorphic to a subgroup of quotient of $\mathbb{Q}$ I've seen this statement written everywhere but without proof and I couldn't prove it (actually its ...
Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G]$ a vector space with basis \$(e_g, g\...