Questions tagged [infinite-groups]

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

327 questions
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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?
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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, ...
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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 ...
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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 ...
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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 ...
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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 $\gamma_i(G) < H$ ?
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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 "...
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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 ...
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 ...
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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 ...
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$G$ 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 ...
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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,...
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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 ...
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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 $G$...
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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 ...
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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 ...
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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 \...
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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?
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\phi(r) > 0$. ...
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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 ...
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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 ...
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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 ...
118 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 ...