Questions tagged [infinite-groups]

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

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If every finite quotient group of a finitely generated linear group G is solvable, then G is solvable

For this question, I was able to show that each finite quotient is polycyclic: Suppose $N$ is a normal subgroup of finite index. Then, all subgroups of $G/N$ are finite, so $G/N$ is Noetherian. A ...
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If every finitely generated subgroup of a linear group G is solvable, then G is solvable

I've been working on this question for a long time now and I still have no idea how to approach it. I tried induction on the dimension of the matrices in G but can't complete the induction step. Any ...
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If ratio: number of integers in between twin primes to number of all integers approaches but never reaches zero, does it imply infinite twin primes? [closed]

Let \xi denote the set of positive integers in between a twin prime. For example, in between 3 and 5 is 4. In between 5 and 7 is 6, and in between 11 and 13 is 12. So 4, 6 and 12 are all in \xi. I ...
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Is every subgroup of infinite Boolean group finite?

Definition (Boolean group): A group $(G,*)$ is said to be Boolean if every non identity element has order $2$ $i.e$ for any $a\in G,o(a)=2$, where $a\neq e$ Now I want to know that given any infinite ...
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There exist infinite solvable $p$-groups with trivial centre. (Use a hint.)

This is Exercise 5.2.11 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. This is marked as being referred to later on in the ...
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The Frattini subgroup of the standard Wreath product of two quasicyclic groups is the group itself.

This is part of Exercise 5.2.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. Here are some previous questions ...
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For any finite group $H$, there's a composition series for $A_{\infty}\times H$

Denote $A_{\infty}=\underset{n\in\mathbb{N}}{\bigcup}A_{n}$. For any finite group $H$, the group $A_{\infty}\times H$ have a composition series. I've shown that $A_{\infty}$ is simple, therefore it's ...
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Every infinite G has a S.T $o(a)=\infty?$ [closed]

Is it true to claim that for every infinite Group G then there is a in G such that $o(a)=\infty?$ I took few examples and this sound correct, but any ideas on how can I prove this in general?
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Isomorphisms for Infinite Direct Products of Groups

Here is a question from Section 2.13 of Herstein's "Topics in Algebra" (2nd edition): If $G_{1}$, $G_{2}$, $G_{3}$ are groups, prove that $(G_{1} \times G_{2}) \times G_{3}$ is isomorphic ...
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Is a subgroup determined by where the generators are in its cosets?

Let $G$ be a finitely generated infinite group and $H$ be a subgroup of finite index. In particular say $G=\langle x_1,x_2,...,x_n\rangle$ and $G:H=\{R_1,...,R_m\}$. Does the distribution of the $x_i$ ...
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Tarski Monsters are not amenable

A Tarski Monster for a prime $p$ is usually defined as an infinite simple group whose proper non-trivial subgroups are cyclic of order order $p$. In "Survey of some results deduced with the help ...
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Every infinite finitely generated group $G$ has an infinite quotient with only finite proper quotients
I am trying to prove this statement: Every finitely generated group $G$ that is infinite has a normal subgroup $K$ such that $G/K$ is infinite and has only finite proper quotients. This is a problem ...
Suppose $G$ is a group and $H$ is a subgroup of $G$. Let's call $T \subset G$ a left/right transversal of $H$ iff it is a system of representatives of left/right cosets of $H$ respectively. Let's call ...