# Questions tagged [infinite-groups]

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

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### Example of a finitely generated metabelian group whose Fitting subgroup is not nilpotent

It is known that the Fitting subgroup of a finitely generated polycyclic-by-finite group is nilpotent, but this statement is not true for the solvable group. It is clear that both Lamplighter groups ...
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### Suppose $H$ is a normal subgroup of $G$ with prime index $p$, does this implies that $H$ contains $G^p$?

The power subgroup is defined as $G^p =\left\{g^p \mid g \in G\right\}$. Question: If $H$ is a normal subgroup of $G$ with prime index $p$, does this imply that $H$ contains $G^p$? This statement is ...
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### Understanding the completed group algebra of a profinite ring.

I am learning about the completed group algebra $k[[G]] = \underset{\underset{N\unlhd_{o}G}{\leftarrow}}{\lim}$ $k[\frac{G}{N}]$ of a profinite group $G$ and a finite field $k$. As far as I ...
1 vote
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### Can we classify the structure of an infinite soluble $p$-group of bounded exponent?

By Baer-Pruffer Theorem, we know that, if $A$ is an abelian $p$-group of bounded exponent, then $A$ is a direct sum of cyclic $p$-groups. Now, assume that $A$ is an infinite soluble $p$-group of ...
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### Proving that a group is infinite and nonabelian

As an exercise I am trying to prove that the group $$G = \langle a,b,c \mid ac = ba, ab=ca, bc=ab\rangle$$ is infinite and non-abelian. Moreover, the author claims that its center has finite index. I ...
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### Can we conclude that an infinite bounded exponent periodic locally nilpotent group $G$ is the direct product of its Sylow $p$-subgroups?

When $G$ is a finite nilpotent group, we know that $G$ is the direct product of its Sylow subgroups. Now, assume that $G$ is an infinite locally nilpotent group. Also, consider that there exists a ...
1 vote
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### Can there be a subgroup $K$ of a group $G$ such that for some $a \in G$, $aK \subseteq Ka$ but $Ka \nsubseteq aK$? [duplicate]

I have a question, in a sense, about how asymmetric left and right cosets can be when dealing with an infinite, non-normal subgroup $K$ of a (non-abelian) group $G$. Specifically, my question is ...
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### If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is abelian [duplicate]

If every proper subgroup of an infinite group $G$ is cyclic, then $G$ is abelian ? I know there is an infinite abelian with this property(every proper subgroup is cyclic). But I was unable to find any ...
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1 vote
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### Do profinite groups admit maximal subgroups

I've been looking into profinite groups, their topological subgroup lattices, etc. I asked the question does every profinite group admit maximal subgroups? I can't find an example of a profinite group ...
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### Basic closed subsets of Stone topological group presented as inverse limit, inner automorphism group of profinite group

Take a profinite group $G=\varprojlim G_\alpha$. We know that the inner automorphism group $\text{Inn}(G)$ of $G$ is profinite since $\text{Inn}(G)=G/Z(G)$, and the quotient of a profinite group by a ...
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