# Questions tagged [infinite-groups]

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

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### 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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### 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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### 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 $6$, $\operatorname{Aut}(S_n) \cong S_n$, $\operatorname{Aut}(D_4) \cong D_4$ ...
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### 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 ...
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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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### 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 ...
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### 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 $p$-...
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### 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 ...
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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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### 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, ...
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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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### 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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### 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 ...
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### 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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### 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 ...
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### 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) ...
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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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### Residually finite nilpotent group

It is known that every finitely generated nilpotent group is residually finite. Why finitely generated hypothesis is essential?
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### 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 ...
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### 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?
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### 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 ...
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### 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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### 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. ...
### 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 $\mathbb{R}/\mathbb{Z}$...