Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [infinite-groups]

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

0
votes
0answers
259 views

How do I show a one-to-one correspondence between a line and a line segment?

Can it be done in the same way as we prove that the number of points on any two finite line segments are equal - geometrically?
1
vote
1answer
153 views

Group of order-automorphisms of the rational numbers

What reference do you recommend for the group $\mathrm{Aut}(\mathbb{Q},<)$ of all order-automorphisms of the rational numbers? Needless to say, this is not about field-automorphisms. Obviously, ...
3
votes
4answers
349 views

Infinite group theory recommendations

What is a good book to start a journey in the field of infinite group theory ? I have already taken a first course in algebra where we studied the most important (finite) algebraic structures and I'm ...
1
vote
1answer
136 views

equinumerous between a finite and infinite set

I need to write a proof that, given A is a finite set and B is an infinite set, show that A is equinumerous to a subset of B. I understand to show this I need to show a bijection. I can get the ...
2
votes
0answers
26 views

Intersection of orbit of prime order element in symplectic algebraic group with maximal subspace subgroup

Suppose $K$ is an algebraically closed field of characteristic $p>0$. Let $G = \mathrm{PSp}(2m,K)$ ($m$ odd) and $H = (\mathrm{Sp}(m-1, K) \times \mathrm{Sp}(m+1, K)) \cap G$ be a maximal ...
1
vote
1answer
26 views

When does $\gamma_i(G/H) = 1$ imply $\gamma_i(G) < H$ ?

Let $\gamma_i$ be the $i$th term in the lower central series for a group $G$ and let $H < G$ be a subgroup of G. When does $\gamma_i(G/H) = 1$ imply $\gamma_i(G) < H$ ?
29
votes
3answers
1k views

Generalisation of integers for infinite length?

As most people know, an integer can only be finite length when expressed in the form of a series of digits in some base. However, real numbers in general can be infinite length, so long as there is a "...
5
votes
2answers
1k views

Probability that 3 points are collinear.

If we select three independent and random points $A$,$B$ and $C$ in a plane, what shall be the probability that they are collinear? Actually, this problem was asked to my friend in an interview, he ...
13
votes
3answers
611 views

Is the unit circle the only infinite compact subgroup of the multiplicative group of non-zero complex numbers?

Let $G$ be an infinite subgroup of $(\mathbb C \setminus \{0\},.)$ such that $G$ is compact as a subset of $\mathbb C$ , then is it true that $G=S^1$ ? I know that $G \subseteq S^1$ ; and since any ...
8
votes
1answer
236 views

How do the subgroups of $(\mathbb C,+)$ which are connected (path connected) in $\mathbb C$ look like?

How do the subgroups of $(\mathbb C,+)$ which are connected in $\mathbb C$ look like ? Can they be characterized in some way ? (like for example , subgroups of $(\mathbb C\setminus \{0\} , .)$ , that ...
4
votes
1answer
117 views

$G$ be a subgroup of $(\mathbb R^2,+)$ such that $\mathbb R^2/G \cong \mathbb Q$ , then is $G$ dense in $\mathbb R^2$?

Let $G$ be a subgroup of $(\mathbb R^2,+)$ such that $\mathbb R^2/G \cong \mathbb Q$ , then is $G$ dense in $\mathbb R^2$ ? If it was $\mathbb R$ instead of $\mathbb R^2$ then I know it would be dense ...
2
votes
1answer
466 views

For an element with infinite order, how many generators does $\left \langle a \right \rangle$ have

Question: Let a be an element of a group and suppose that a has infinite order. How many generators does $\left \langle a \right \rangle$ have? Following from the hypothesis that a has infinite order,...
3
votes
1answer
77 views

Is the intersection of a descending chain of group retracts a retract?

Let $G$ be a group. A subgroup $H\subset G$ is said to be a retract if there exists surjective homomorphism $\pi:G\to H$ s.t. $\pi\circ\iota=Id_{H}$ with $\iota$ being the inclusion of $H$ in $G$. So ...
1
vote
2answers
87 views

Construct a strictly descending infinite sequence of non-commutative subgroups

Let $\Omega$ denote the set of integer tuples $(a,b)$ where $a + b$ is odd. We de fine two permutations of $\Omega$ as: $ \sigma : (a,b) \mapsto (a+1,b-1)$ and $ \delta: (a,b) \mapsto (b,a)$ Let $G$...
1
vote
1answer
174 views

The closure of the set of all permutations with finite support in $Sym( \mathbb{N})$

https://www.math.tu-dresden.de/~bodirsky/lehre/alg-strukturen-ss-2016/script.pdf Consider $Sym( \mathbb{N})$ as a topological group. Let $P$ be the set of permutations $f$ of $\mathbb{N}$ that have ...
1
vote
1answer
47 views

Conjugacy of maximal subgroups

This is inspired by my earlier duplicate question seen here: Is conjugation in infinite groups well behaved? The answer to that question is no. This slight tweak might change the result, but I don't ...
1
vote
0answers
39 views

Is conjugation in infinite groups well behaved? [duplicate]

A question from earlier today made me start to think of this question. Let $G$ be an infinite group with infinite subgroup $H$. Can the condition below ever hold? \begin{equation*} \exists g\in G \...
2
votes
0answers
63 views

Existence of a specific group(Group Theory)

Does there exist an infinite group G, such that for any two non trivial subgroups H and K, H is not a subset of K and K is not a subset of H?
13
votes
1answer
163 views

Injecting a group into itself

Let $G$ be a group with elements $g_1, g_2\in G$ and injective endomorphisms $\phi_1, \phi_2$ s.t. $\phi_1 (g_1)=g_2$ and $\phi_2(g_2)=g_1$. Does this imply there is an automorphism $\psi$ with $\...
1
vote
1answer
233 views

An infinite group $G$ always has a countably infinite subgroup.

I tried to find out whether the following good-looking statement about groups is true. "An infinite group $G$ always has a countably infinite subgroup." I faced this question while pondering upon the ...
1
vote
3answers
3k views

What are some examples of infinite nonabelian groups?

I can only think of the general linear groups GLn(R). What are some other examples, if any?
4
votes
2answers
60 views

Question about an equivalent definition of a simple group

A basic consequence of the first isomorphism theorem is that a finite group G is simple if and only if its only homomorphic images are G and the trivial group (up to isomorphism). However, I'm not ...
0
votes
2answers
402 views

Giving examples of denumerable sets [closed]

My question reads: Give an example of denumerable sets A and B, neither of which is a subset of the other, such that (a) A ∩ B is denumerable (b) A-B is denumerable. I am not sure if I would ...
0
votes
1answer
430 views

proving infinite set from a finite and infinite set [duplicate]

My question reads: If A is finite and B is infinite, then B-A is infinite.
1
vote
0answers
64 views

On understanding the structure of pure subgroups of a $p$-primary group.

For any abelian group $G$ ad any $n\in\mathbb N$, let $G[n]$ denote the subgroup $\{g\in G\mid ng=0\}$. Now suppose $G$ is a $p$-primary group for some prime number $p$ and $H$ $H^\prime$ two pure ...
3
votes
1answer
120 views

Is the action of $SO(3, \mathbb R)$ on $S^2:=\{(x,y,z) \in \mathbb R^3 |x^2+y^2+z^2=1\}$ defined as $(A,x) \to Ax$ transitive?

Consider the action of $SO(3, \mathbb R)$ on $S^2:=\{(x,y,z) \in \mathbb R^3 |x^2+y^2+z^2=1\}$ as $(A,x) \to Ax$ . Is this action transitive ? i.e. is it true that for every $x,y \in S^2$ , $\exists A ...
-3
votes
1answer
66 views

elements of finite order in G [closed]

Let G be the quotient group (ℤ×ℤ)/⟨2,4⟩ What are the elements of finite order and how do i find how many are there? Does this make G cyclic? I am stuck on how to start this
2
votes
0answers
91 views

A class of subsets of the sums of $\mathbb{Z}_2$

Consider the additive group $G=\sum_{i\in I}\mathbb{Z}_2$, and for every $A\subseteq G$ put $A-A=\{a_1-a_2: a_1,a_2\in A\}$, $$ S_A:=\{B\subseteq G: (A-A)\cap(B-B)=\{0\} , (A-A)+B=G\} $$ Now, is it ...
2
votes
0answers
150 views

Infinite-dimensional irreducible representations of countable abelian groups

Doesn't seem exactly a research level question to me, so posting here instead of MO. I seem to know a proof of the (counter-intuitive, in my opinion) fact that a countable abelian group may not have ...
1
vote
0answers
60 views

Group with a special generating set

Let $G$ be a group such that: 1) $G$ can be generated by a finite set of elements of infinite order. 2) In $G$ there is a subgroup $H$ such that $H$ is a free group and its index $|G:H|$ in $G$ is ...
4
votes
1answer
136 views

Looking for a proof that if a normal subgroup of $Sym(\mathbb N)$ has an element of infinite order then it is the whole group

I know that $Sym (\mathbb N)$ , the group of all permutations on $\mathbb N$ , has exactly two non-trivial normal subgroups and both of these subgroups are torsion , so it follows that if a normal ...
2
votes
1answer
182 views

Is this an isomorphism possible?

I am working on the following homework problem: Let $\phi$ be an isomorphism from $\mathbb{R}^*$ to $\mathbb{R}^*$ (nonzero reals under multiplication). Show that if $r>0$, then $\phi(r) > 0$. ...
1
vote
3answers
365 views

Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups under the axiom of choice?

I know that there was also a similar question asked before but I don't really understand the solution. There was said: "Assuming the axiom of choice, yes. Observe that both these abelian groups ...
4
votes
1answer
212 views

Groups with several direct product decompositions

The Krull—Schmidt theorem states that every group with ACC and DCC on its normal subgroups has only one decomposition into the product of directly indecomposable groups (in particular, this holds for ...
0
votes
1answer
50 views

What is wrong with this demonstration regarding the sum of all naturals? [duplicate]

If this is correct, then the demonstration below must have a fault, but I can't find it. Assuming $1+2+3+\cdots = -\frac{1}{12}$ (1) is true. Adding $0$ on both sides: $0 + 1+2+3+\cdots = 0 -\...
0
votes
1answer
33 views

Equivalence relation on cosets

In lecture was defined for groups H and its cosets: $ a \sim_L b $, if $ aH = bH$ and $ a \sim_R b $, if $ Ha = Hb $ For finite groups one can find $a,b \in H$ such that $a \neq b$, $aH = bH \neq ...
2
votes
4answers
514 views

Subgroup of infinite group

Are there any infinite order groups such that they have finite order non trivial subgroup? The simplest subgroup would be one where the element is its own inverse other than identity. Next case is if ...
7
votes
1answer
118 views

Are there non-abelian groups with the property $|AB|=|BA|$?

Regarding to the question "Is there any non-abelian group with the property $AB=BA$?", now it is important for us to know that: (a) Is there any finite (resp. infinite) non-abelian group of order $\...
5
votes
1answer
179 views

Is there any non-abelian group with the property $AB=BA$?

Is there any finite (resp. infinite) non-abelian group of order $\geq 8$ such that $AB=BA$ for all subsets $A, B$ with $|A|\geq 3$ and $|B|\geq 3$? ($AB=\{ab: a\in A, b\in B\}$)
0
votes
2answers
461 views

Number of homomorphisms from a finite group to an infinite group

I am starting to self-study abstract algebra, and came across, in my humble opinion, a fairly difficult question. More specifically, the question I was trying to answer is along the lines of: "Let $G$...
1
vote
1answer
49 views

Is the group $S^1:=\{z\in \mathbb C :|z|=1\}$ isomorphic with $S^1\times S^1$ ?

Is the group $S^1:=\{z\in \mathbb C :|z|=1\}$ isomorphic with $S^1\times S^1$ ? I know that $\mathbb C^* \cong S^1 \cong \mathbb R/\mathbb Z \cong \mathbb C^*/\mathbb R^+$ , but I cannot make any ...
5
votes
1answer
2k views

The Perfect Sharing Algorithm (ABBABAAB…)

Less of a question and more of an exercise, it has to do with something I found while doing some programming and being unable to find things. Basically I wanted a formula for the perfect sharing ...
1
vote
1answer
30 views

Is the torsion set of the group of permutations on $\mathbb N$ closed under composition?

Let $S(\mathbb N)$ be the permutation group over $\mathbb N$ , then is it true that there exist elements $f,g\in S(\mathbb N)$ of finite order such that $f\circ g$ is not of finite order ?
0
votes
1answer
341 views

Subgroup of the infinite dihedral group D_inf that is isomorphic to D_inf

We define $D_{\infty} := \langle r, s \ | \ s^2 = e, srs = r^{-1} \rangle = \lbrace ..., r^{-2}, r^{-1}, r, r^2, ..., e, ..., r^{-2}s, r^{-1}s, s, rs, r^2s, ... \rbrace$. I am trying to find a ...
0
votes
0answers
68 views

How to show $\langle x,y \mid x^3=y^3=(xy)^3=1\rangle$ is presentation of an infinite group? [duplicate]

How to show $\langle x,y \mid x^3=y^3=(xy)^3=1 \rangle$ is presentation of an infinite group? This is a result in textbook, but I do not understand the rationale.
1
vote
1answer
89 views

Let $H$ be the set of all $f\in \mathcal{S}(A)$ such that $f(x)=x$ for all but a finite number of elements $x$ of $A$. Prove $H\leq\mathcal{S}(A)$.

Let $A$ be an infinite set, and let $H$ be the set of all $f\in \mathcal{S}(A)$ such that $f(x)=x$ for all but a finite number of elements $x$ of $A$. Prove that $H$ is a subgroup of $\mathcal{S}(A)$. ...
6
votes
0answers
145 views

$H$ is a normal subgroup (containing an element of infinite order) of the permutation group over positive integers , then is $H$ the whole group?

Let $S(\mathbb N)$ be the set of all bijections on $\mathbb N$ ( the set of positive integers ) endowed with the natural group structure of function composition . If $H$ is a normal subgroup of $S(\...
3
votes
2answers
72 views

$H,K$ be subgroups of an infinite abelian group $G$ such that $K \subseteq H$ and $G \cong K$ , then is it true that $G\cong H$ ?

Let $H,K$ be subgroups of an infinite abelian group $G$ such that $K \subseteq H$ and $G \cong K$ , then is it true that $G\cong H$ ? ( see related $C$ be a subring of $B$ which is again a subring of ...
1
vote
2answers
151 views

Find an example infinite group

Find an example of an infinite group and a set of elements $\{g_{1},g_{2},\dots,g_{n},\dots\}$ of finite order with the property that if $$S_n=\cfrac{o(g_{1})+o(g_{2})+\cdots+o(g_{n})}{n}$$ then $\...
1
vote
1answer
38 views

The equality $G=A^{-1}A\cup A\cup A^{-1}$ [closed]

Let $G$ be a finite (resp. an infinite) group, $A\subseteq G$ and put $A^{-1}:=\{ a^{-1}:a\in A\}$. Is it true that $G=A^{-1}A\cup A\cup A^{-1}$ implies $G=A^{-1}A$?