Questions tagged [infinite-groups]

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

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4
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1answer
59 views

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 ...
1
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0answers
48 views

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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0answers
28 views

Order of completely reducible group action

Let $\mathbb{K}$ be an algebraically closed field and $V$ an n-dimensional $\mathbb{K}-$vector space. Suppose $G \leq GL(V)$ is completely reducible and that for some $d \in \mathbb{N}$ we have $ g^d =...
2
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2answers
56 views

Probability of picking a number from a set with a uncountably infinite number of positive numbers and a countably infinite number of negative numbers?

Sorry if this is a stupid question. I couldn't see anything else quite the same as this, although I found some similar questions. Please remove this if it is a duplicate. (I hardly understand sets so ...
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2answers
123 views

Let $G$ be an infinite group and let $H$ be a finite subgroup. Does there exist a finite index normal subgroup of $G$ which contains $H$?

Let $G$ be an infinite group, and let $H$ be a finite subgroup. Must there exist some finite index proper normal subgroup of $G$ which contains $H$? If this does not hold for general infinite groups $...
0
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1answer
82 views

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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2answers
69 views

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 ...
1
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1answer
78 views

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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0answers
44 views

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 ...
0
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1answer
31 views

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 ...
0
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1answer
38 views

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?
3
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1answer
135 views

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 ...
6
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1answer
276 views

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 $x_i$ ...
1
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1answer
33 views

Group G Preserving a Flag is Solvable

Suppose $V$ is a vector space of dimension $n$ over the field $\mathbb{K}$ and $$V_0 \subsetneq V_1 \subsetneq ...\subsetneq V_k =V $$ is a flag (not necessarily complete) preserved by a subgroup $G \...
5
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3answers
128 views

Infinite abelian group whose all nontrivial subroups have finite index

D.J.S. Robinson, A Course in the Theory of Groups, 2d edition, exercise 4.1.3, p. 98, asks for a proof of the following statement : Statement 1. If $G$ is an infinite abelian group all of whose proper ...
1
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1answer
49 views

Existence of deep enough open subgroups in profinite groups

Let $G$ be an infinite profinite group and $\{U_i\}_{i \in I}$ be any family of open subgroups of $G$. Is it possible to choose a family of open subgroups $V_i < U_i$ with the property that, for ...
3
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1answer
39 views

Prove that every proper subgroup of the group of all $2^n$-th roots of unity is finite.

Find an abelian infinite group such that every proper subgroup is finite I've seen the answer here but I'm really struggling to understand and prove it. What I understand is that if I suppose that $H$ ...
3
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2answers
119 views

Abelian group $G$ such that the infinite-order elements form a subgroup with the identity.

Let $G$ be an abelian group. If $\{g \in G \mid g=e \text{ or }g \text{ has infinite order}\}$ is a subgroup of $G$, what can we say about the order of the elements of $G$? My observations: It is ...
1
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2answers
62 views

Group generated by matrices with integers off the diagonal

I would like to know the subgroup generated by the following type of matrices. $$A=\left(\begin{matrix}1&a\\ 0&1\end{matrix}\right)$$ and $$B=\left(\begin{matrix}1&0\\ b&1\end{matrix}\...
4
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1answer
49 views

Prove that if a subgroup $H$ generated by $x$ is of infinity order, then $H=\langle x^a\rangle$ iff $a = ±1$

I am trying to prove a proposition in Dummit and Foote book which says: Let $H = \langle x \rangle$. Assume $|x| = \infty$.Then $H = \langle x^a \rangle$ if and only if $𝑎=±1$. My attempt was: If $𝑎=...
3
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1answer
86 views

Why is $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$ not isomorphic to $\mathbb{Z}_2 \times \bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$

I was wondering if someone could offer a hint on how to show that $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$ is not isomorphic to $\mathbb{Z}_2 \times \bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$? Here ...
1
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1answer
78 views

Involution elements in infinite groups

In this, involutions in a finite group are either conjugate or have an involution centralizing both of them. I wonder if there are similar results for an infinite group. I think and look for it but I ...
4
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1answer
158 views

Just infinite group

A group is said to be just infinite if it is infinite and every proper quotient is finite. In this paper http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160210604/pdf I read that infinite dihedral ...
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2answers
218 views

Is there a surjective homomorphism from $(\Bbb{Q},+)$ to $(\Bbb{Z},+)$?

Consider the groups $G = (\Bbb{Q},+)$ and $H = (\Bbb{Z},+)$. Is there a surjective homomorphism from $G$ to $H$? If not, how can I prove there isn't? I considered a homomorphism that rounds up or down ...
1
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0answers
67 views

Show that a group, given by a presentation, is countable & reduced with a nontrivial element of infinite height

This is part of Exercise 4.3.7 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. [NB: I did not include the combinatorial-group-...
0
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2answers
67 views

Isomorphism between $\mathbb{R}^*$ and $\mathbb{R}\times \mathbb{Z}_2$

I want to show that $\mathbb{R}^*\cong \mathbb{R}\times \mathbb{Z}_2$. I don't know where to start, but I know that $f(1)=(0,\overline{0})$ because isomorphisms take the neutral element to the ...
0
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1answer
136 views

Which groups have non-trivial cyclic quotient groups?

Under what conditions can it be determined whether or not any group G has some subgroup N such that the quotient group G/N is a cyclic group? In other words, how can it be determined whether or not a ...
0
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0answers
76 views

Using a particular theorem, characterise abelian groups which have only finitely many elements of each order (including $\infty$)

This is part of Exercise 4.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. Here is the previous part: An ...
5
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1answer
150 views

An abelian $p$-group has finitely many elements of each order iff it satisfies ${\rm min}$

This is part of Exercise 4.3.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. (NB: I have left out the modules ...
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0answers
56 views

Proof that $(\mathbb{Z}[x], +) \times \mathbb{Z} \times \mathbb{Z} \cong (\mathbb{Z}[x], +) \times \mathbb{Z}$

In this post on Art Of Problem Solving, it is claimed that the following group isomorphism holds: $$(\mathbb{Z}[x], +) \times \mathbb{Z} \times \mathbb{Z} \cong (\mathbb{Z}[x], +) \times \mathbb{Z}$$ ...
0
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1answer
64 views

Evaluate the limit of $\lim_{x \to \infty} \sqrt{x} (\sqrt {x} - \sqrt {x -a})$ [closed]

The question is in my book and I searched for it and got this solution. I have reached upto $$\lim_{x \to \infty} \sqrt{x} \frac { a } {(\sqrt{x} + \sqrt{x−a})}$$ However, I couldn't complete further. ...
1
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1answer
59 views

For $H\le G$, abelian $G$, show that $r_0(H)+r_0(G/H)=r_0(G)$, where $r_0(K)$ is the torsionfree rank of $K$

This is Exercise 4.2.7(i) of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. I've asked two previous questions on based on earlier ...
2
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1answer
81 views

If $G$ is abelian, show the Prüfer rank $r(G)$ is finite iff $\max\{ d(H)\}$ is finite, for $d(H)$ the min number of generators of f.g. $H\le G$

This is part of Exercise 4.2.2 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The Details: On page 98 to 99, ...
3
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1answer
53 views

Supremum of the cardinalities of the minimum no. of generators among finite subgroups of $GL(n,F)$

For a finite group $G$, set $\mu(G):=\min\{|S|: G=\langle S\rangle\}$. Given an infinite field $F$ and an integer $n\ge 1$, set $$\mu_{n,F}:=\sup\{\mu(G): G \le GL(n,F), |G| \text{ is finite}\}.$$ $$\...
3
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2answers
66 views

Can every countable locally-finite group be embedded into $S_\infty$?

Let's define $S_\infty$ as the group of all permutations of $\mathbb{N}$ with finite support. It is not hard to see, that every finite group can be embedded in $S_\infty$. That is because any finite ...
8
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1answer
141 views

$\mathbb{K}^{\times} \times \mathbb{Z}$ is not group isomorphic to $\mathbb{K}^{\times}$

Let $\mathbb{K}$ be a field with infinite cardinality and $\mathbb{K}^\times$ its group of units under multiplication (i.e. all elements except $0$). I want to determine if $\mathbb{K}^{\times} \times ...
2
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0answers
101 views

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 ...
4
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1answer
125 views

Quotient of $\mathbb{R^*}$ by cyclic group product

For any $k \in \mathbb{R^*}, \langle k\rangle$ is a normal subgroup. Consider the case $k \neq 1, k>0$. Then, via the surjective homomorphism $$\varphi: \mathbb{R^*} \to \mathbb{T}\times\{\pm1\}, \...
3
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1answer
230 views

Infinite subgroups of SO(3)

The classification of finite subgroups of SO(3) is well-known: we have the cyclic groups, the dihedral groups, and the symmetries of the Platonic solids. Is there an analogous result for the infinite ...
0
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0answers
48 views

On a result of Baer on the group $\prod_{n=1}^{\infty} \mathbb{Z}$

I was looking proof of a result of Baer for his result: the infinite abelian group $\prod_{i=1}^{\infty} \mathbb{Z}$ is not free abelian group. I was looking proof in Rotman's Introduction to ...
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0answers
20 views

Equivalent definition of nilpotency for infinite groups

Let $G$ be a finite group. TFAE: (1) $G$ is nilpotent. (2) $N_{G}(H) > H$ for all subgroups $H< G$. (3) Every maximal subgroup of $G$ is a normal subgroup. The proof of these equivalences seems ...
1
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2answers
50 views

If $S$ is all finitary permutations of $\Bbb N$ and $A$ is the alternating subgroup, prove $1\lhd A\lhd S$ is the only composition series of $S$

This is Exercise 3.2.2 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. Previous exercises of mine from the book include: ...
2
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1answer
52 views

Ends of a group and Spherical growth function

I am trying to get some intuition about a spherical growth function of a group, using the notation from A. Mann $s_G(m)=\sum_{n=0}^m a_G(n)$, where $s_G(m)$ is the cumulative growth function and $a_G(...
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1answer
58 views

If $G$ is an infinite group, what can you say about the number of elements of order $n$ in the group? [closed]

I just started with group theory. I know that for any finite group $G$, the number of elements of order $n$ in group $G$ will be multiple of $\phi(n)$ where $\phi$ is the Euler phi function. But what ...
-1
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1answer
109 views

How to find a group with generators and relations fulfilling several properties? [closed]

I would like to ask the following question: Let $G:= \langle x,y \mid \text{some relations} \rangle$. Is it possible to find a group $G$ as above fulfilling the following criteria simultaneously? ...
2
votes
1answer
224 views

A group isomorphism between $\mathbb{Q/Z}$ and $\mathbb{Q/2Z}$

Question: Prove that $\mathbb{(Q/Z, +)}\cong\mathbb{(Q/2Z, +)}$ My attempt To prove they are isomorphic I need to define a map from $\mathbb{Q/Z}$ to $\mathbb{Q/2Z}$ which is bijective and preserve ...
0
votes
1answer
85 views

Show that every proper subgroup is finite

For any prime $p$, show that every proper subgroup of the Prüfer $p$-group $(\{a/p^n+\mathbb{Z}:a, n \in \mathbb{Z}, n\geq0\}) $ is finite. Given a subgroup $H$, I suppose this should have order $p^k$...
0
votes
1answer
65 views

Are these two infinite groups isomorphic? [duplicate]

Question. Prove or disprove that $G_1$ and $G_2$ are isomorphic. $$G_1=\mathbb Z_5 \times \mathbb Z_{5^2}\times \mathbb Z_{5^3}\times \mathbb Z_{5^4} \times \cdots$$ $$G_2= \mathbb Z_{5^2}\times \...
1
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0answers
82 views

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 ...
4
votes
1answer
102 views

Does there always exist a double transversal?

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 ...

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