Questions tagged [infinite-groups]

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

1
vote
2answers
124 views

$\mathbb{Z}\times \mathbb{Z}_{2}$ is a cyclic group?

I think that $\mathbb{Z}\times \mathbb{Z}_{2}$ isn't a cyclic group becuase we don't have any $(a,b)\in \mathbb{Z}\times \mathbb{Z}_{2}$ that can create the group $\mathbb{Z}\times \mathbb{Z}_{2}$. I'...
9
votes
3answers
235 views

Can $\mathbb{R}$ be written as an ascending union of proper additive subgroups?

Can the group $\mathbb{R}$ be written as countable ascending union of proper subgroups? (i.e. does there exists a series of proper subgroups $H_1\leq H_2\leq \cdots $ such that $\cup {H_i}=\mathbb{R}$?...
0
votes
2answers
129 views

Question about Abelian group proof

I prove that if $G$ is Abelian group so if $a,b\in G$ has a finite order so $ab$ has a finite order to.. (Maybe later I'll upload here my proof to see of she is correct....) Now, I have to show that ...
0
votes
2answers
77 views

find a generator for the group $G =\{ f(x) = x+n\mid n\in \Bbb Z \}$ with the group operation being composition.

Another question from 'A book of Abstract Algebra' by Pinter. For each $n\in \Bbb Z$ define $f_n = x+n$. Then $f_n\in S_{\Bbb R}$, the symmetric set on $\Bbb R$. The group operation being composition....
6
votes
1answer
214 views

Groups with 3 conjugacy classes and finite exponent

I have seen the question on groups with two conjugacy classes, and I proved to myself that such a group must be torsion-free (if it isn't the cyclic group of order 2), but what about a group with ...
1
vote
0answers
38 views

Cyclic Groups of Infinite and Finite order [duplicate]

If a cyclic group has an element of infinite order , how many elements of finite order does it have ?
2
votes
3answers
101 views

Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$

Define $H = \{(a, 0): a\in \mathbb{R}\}$. Without using the fundamental homomorphism theorem, how would we know what $\mathbb{R}^2/H$ is? The quotient group is $\{H, (x, y) + H, (x_2, y_2) + H, \dots ...
3
votes
0answers
399 views

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. ...
1
vote
1answer
72 views

Can an infinite permutatation be decomposed into finite number of infinite cycles?

Let $\sigma \in Perm(\mathbb{N})$ the set of permutations on the naturals. Then can $\sigma$ be written as a finite composition of possibly infinite disjoint cycles?
2
votes
1answer
81 views

Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?

Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$? $G_\alpha$ is the set of elements of $G$ ...
2
votes
2answers
375 views

Residually finite nilpotent group

It is known that every finitely generated nilpotent group is residually finite. Why finitely generated hypothesis is essential?
2
votes
2answers
2k views

Example of an infinite group where every element except identity has order 2

Find an infinite group, in which every element g not equal identity (e) has order 2 Does this question mean this: the group that fail condition (2) which is no inverse and also that group must have ...
6
votes
0answers
2k views

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 ...
2
votes
3answers
909 views

Does every infinite group has infinite cyclic subgroup?

I need a help with determining if this statement is true or not. Does every infinite group has infinite cyclic subgroup?
4
votes
2answers
692 views

Finitely presented Group with less relations than Generators.

I some how feel that, any finitely presented group with less relations than Generators has to be an infinite Group. In One of my Questions, In an answer I have seen a group which is finitely ...
9
votes
4answers
6k views

Torsion Subgroup (Just a set) for an abelian (non abelian) group.

Let $G$ be an abelian Group. Question is to prove that $T(G)=\{g\in G : |g|<\infty \}$ is a subgroup of G. I tried in following way: let $g_1,g_2\in T(G)$ say, $|g_1|=n_1$ and $|g_2|=n_2$; Now, ...
1
vote
0answers
113 views

Problem on non abelian tensor product of groups

Let $G$ be group and $x\otimes[y,z] =([x,y]\otimes z)^2$, for any $x,y,z\in G$. Then prove $$[x,y]\otimes y=1_{\otimes},$$ for any $x,y\in G$. This is a part of Corollary 4 in page 82 in the article ...
1
vote
0answers
88 views

How to directly prove an identity on infinite series

$$ \begin{array}{c} \sum\limits_{l=0}^{\infty }C_{m+l}^{l}\frac{z^{x+l}-2^{-x-l}}{x+l}% =\sum\limits_{l=0,l\neq m}^{\infty }C_{l-x}^{l}\frac{2^{m-l}-\left( 1-z\right) ^{l-m}}{l-m}-C_{m-x}^{m}\ln (2-2z)...
10
votes
3answers
275 views

Classify all groups containing isomorphic copy of $\mathbb{Z}$ of index $2$.

I have the following question: Classify all groups $G$ containing an isomorphic copy of $\mathbb{Z}$ such that the copy has index $2$ in $G$ There are some candidates: $\mathbb{Z}\times\mathbb{Z}/...
8
votes
2answers
605 views

If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$? [duplicate]

If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$ ? This means, $gHg^{-1}$ is a proper subgroup of $H$. We know that $H \cong gHg^{-1}$, so if $H$ is ...
11
votes
6answers
1k views

Does every infinite group have a maximal subgroup?

$G$ is an infinite group. Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ? Is it possible that there exists such series $H_1 < H_2 < H_3 <\cdots &...
26
votes
2answers
4k views

countable group, uncountably many distinct subgroup?

I need to know whether the following statement is true or false? Every countable group $G$ has only countably many distinct subgroups. I have not gotten any counter example to disprove the statement ...
6
votes
1answer
430 views

An infinite $p$-group may not be nilpotent

It is well-known fact that every finite $p$-group $G$ is nilpotent. I am asking to have a counter example when $G$ is infinite $p$-group. Thanks.
3
votes
2answers
172 views

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?
3
votes
0answers
259 views

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 ...
5
votes
1answer
767 views

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 ...
7
votes
1answer
916 views

What is $\operatorname{Aut}(\mathbb{R},+)$?

I was solving some exercises about automorphisms. I was able to show that $\operatorname{Aut}(\mathbb{Q},+)$ is isomorphic to $\mathbb{Q}^{\times}$. The isomorphism is given by $\Psi(f)=f(1)$, but ...