Questions tagged [infinite-groups]

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

66 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
8
votes
0answers
77 views

For general $n \in \Bbb N$ , how to determine all groups (both finite and infinite) having exactly $n$ conjugacy classes?

I was trying to solve the following problem : Determine all the finite groups having exactly 3 conjugacy classes. My attempt: Among all abelian groups $(\Bbb Z_3,+)$ only satisfies the property , ...
7
votes
0answers
199 views

Is $Z(\Sigma) \cong E$?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
7
votes
0answers
116 views

On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
6
votes
0answers
146 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(\...
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 ...
5
votes
0answers
53 views

Does there exist an Artinian verbally simple group, which is not characteristically simple?

Does there exist an Artinian verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally ...
4
votes
0answers
49 views

Conjugacy class with $1$ element in an Infinite Simple Group

We would like to show that if $G$ is an infinite simple group, then the only conjugacy class of exactly one element is $\{1_G\}$. My thoughts: We want to proove that if $|\mathrm{orb}(x)|=1\iff \...
4
votes
0answers
188 views

Any divisible subgroup of $G$ splits with internal direct product.

The question is as follows: Let $G$ be an Abelian group and suppose that $A$ is a divisible subgroup of finite index. Show that $G = A \dot{\times} B$ for some $B \leq G$. $\textbf{Some definitions ...
4
votes
0answers
122 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
3
votes
0answers
63 views

Automorphism group of $(\Bbb Z\times\Bbb Z) \rtimes \Bbb Z$

Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Lets say the generators of ...
3
votes
0answers
47 views

$\mathbb{Z} $-linear reidually solvable groups

Is there a known important class of $\mathbb{Z} $-linear reidually solvable groups, except for free groups? It is just a rough question and I'm not an expert on this subject. I read some books on ...
3
votes
0answers
593 views

The group of pure/hereditary sets

From wikipedia: In set theory, a hereditary set (or pure set) is a set all of whose elements are hereditary sets. That is, all elements of the set are themselves sets, as are all elements of ...
3
votes
0answers
234 views

Infinite Groups with Finitely many Conjugacy Classes

If $G$ is an infinite group with finitely many conjugacy classes, what can be said about $G$? (Should $G$ be simple/ solvable/....?) For $n\geq 2$, does there exists a (infinite) group $G$ with ...
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. ...
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 ...
2
votes
0answers
50 views

Abelian groups having no finite subgroup

My question is: Are there currently any classification (up to isomorphism) of infinite abelian groups having no non-trivial finite subgroup? It seems many have asked whether an infinite group can ...
2
votes
0answers
52 views

How to enumerate Von Dyck groups?

I would appreciate an algorithm to list all elements of a given Von Dyck group $D(p,q,r)$, each once, in a format that would allow me to find compositions and inversions within that list. ...
2
votes
0answers
32 views

Infinite Abelian Groups: Unnecessary Hypothesis in Kaplansky's Lemma 7

Looking for a validation of a proof of a stronger statement. The essential argument that needs review is the final paragraph. This would obviate the need for a much more complicated argument I gave ...
2
votes
0answers
26 views

Index of a common normal core

Let $A, B, C$ be infinite groups and suppose that there are injective homomorphisms $\iota_A \colon C \to A$ and $\iota_B \colon C \to B$ such that $|A : \iota_A(C)|, |B: \iota_B(C)| < \infty$. ...
2
votes
0answers
34 views

If $A$ is a free abelian normal subgroup, and $K'$ has a unipotent image then $C_A(K')$ is non-trivial

Here's the context of the question: Let $G$ be an infinite polycylic group. It is known that there is $A \triangleleft G$ with $A \cong \mathbb{Z}^d$ for $d >0$. Define a homomorphism $\phi: G \...
2
votes
0answers
30 views

f $G$ is generated by 2 elements, and has a free subgroup $L$ of rank $n+1$, where $|G:L|=n$, then $G$ is free of rank $2$.

A question from a past exam paper I am trying to solve: If $G$ is generated by 2 elements, and has a free subgroup $L$ of rank $n+1$, where $|G:L|=n$, then $G$ is free of rank $2$. I am not sure ...
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 ...
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?
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
152 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 ...
2
votes
0answers
36 views

Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...
2
votes
0answers
188 views

Bijection between the set of irrational numbers from 0 to 1 and the set of infinite integers? (Actually, 10-adic integers)

It seems to me we can assign to every irrational from 0 to 1 an infinite whole number in the following way: $$ 0.a_1a_2a_3a_4... \rightarrow ...a_4a_3a_2a_1 $$ For example: $$ 0.14159265... \...
2
votes
0answers
145 views

Index of Maximal Subgroup

My question is about the index of a maximal subgroup of a p-group. If G is a finite p-group and M a maximal subgroup of G, the we know the index of M in G is p. Do you think the result still holds ...
1
vote
0answers
81 views

Does $\Sigma$ generate the variety of all groups?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
1
vote
0answers
45 views

Correcting Proof of Lemma 10 in Kaplansky's *Infinite Abelian Groups*

Lemma 10 in Kaplansky's Infinite Abelian Groups states: Let $G$ be a primary group, $H$ a pure subgroup, $x$ an element of order $p$ not in $H$. Suppose that $h(x)=r<\infty$, and suppose ...
1
vote
0answers
122 views

Torsion-free abelian groups of finite rank and a subgroup of finite index (Fuchs' problem) - self study

I'm trying to solve the following exercise (Fuchs, "Infinite Abelian Groups", Vol. $2$, pp. $153$, Ex.$5$): Let $A$ be a torsion-free abelian group of finite rank. If $\phi$ is an isomorphism of $...
1
vote
0answers
60 views

Classifying elements of finite and infinite order in $GL_{2}(\mathbb{R})$

A problem on a recent assignment defined the Torsion subset $F(G)$ of a Group $G$ as the set of elements of G of finite order. It then asked to prove that $F(GL_2(\mathbb{R}))$ is not a subgroup of $...
1
vote
1answer
56 views

About infinite chains of subgroups containing an infinite subgroup of a locally finite group

Let $G$ be an infinite locally finite (non-solvable) group, let $\{H_i\}_{i\in\mathbb{N}}$ be a strictly totally ordered family of subgroups of $G$ and let $H$ be the intersection of that family. If $...
1
vote
0answers
26 views

Can the triangularization of a linear group preserve a given good feature of an element?

Let $k$ be an algebraic-closed field and let $G$ be a soluble subgroup of $GL_n(k)$. By a result of Mal'cev (Bertram A.F. Wehrfritz - Infinite Linear Groups, Theorem 3.6), $G$ contains a normal ...
1
vote
0answers
30 views

Subgroup of index p in an infinite p-group?

Does an abelian infinite $p$-group always contain a subgroup of index $p$ ? Thanks.
1
vote
0answers
69 views

Does there exist such a group?

First Question: I was asked, "Find a group that has a subgroup of order $n$ for each positive integer $n$. I considered the nonzero complex numbers under multiplication, and noticed that the roots of ...
1
vote
0answers
51 views

Ulm Invariants of a reduced abelian primary group

Let $G$ be a reduced abelian primary group of lenght $\lambda$, and let $\alpha$ and $\beta$ with $\beta$ a limit ordinal and $\alpha < \beta \leq \lambda$. Show that the Ulm Invariants of $G_{\...
1
vote
0answers
44 views

Certain property of free products of groups

Let $G = A\ast B$ be a free product of groups $A\ne1$ and $B\ne1$ without the elements of order $2$. Suppose that $f\in G$ and $f\notin A^g,B^g$ for any $g\in G$ and let $\left \langle \left \langle f ...
1
vote
1answer
156 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, ...
1
vote
1answer
137 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 ...
1
vote
0answers
65 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 ...
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 ...
1
vote
0answers
120 views

If the group is infinite , what inference should I make about the number of nonidentity elements that satisfy the equation $x^5=e$?

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5=e$ is a multiple of 4. If the stipulation that the group be finite is omitted, what can you say about the ...
1
vote
0answers
80 views

On centers of infinite $p$-groups and nilpotent groups

If $G$ is a finite $p$-group, then its center is non-trivial, which forces that $G$ must be nilpotent. Consider infinite $p$-groups, i.e. infinite groups in which order of every element is some power ...
1
vote
1answer
127 views

Infinite nilpotent groups

We know that all subgoups in a finite nilpotent groups are subnormal subgroup. Is there exists an infinite nilpotent group, whose all subgroups are not subnormal subgroups?
1
vote
0answers
50 views

Exponential map from loop algebras to loop groups

In Terry Gannon's book, 'Moonshine beyond the Monster', on page 206, he mentions that for a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, the exponential map from the loop algebra $L\mathfrak{...
1
vote
0answers
181 views

example of p-group with infinite Frattini subgroup

Is there an infinite $p$-group $G$ with infinite Frattini subgroup $\Phi(G)\not = G$? In "Subgroups of Teichmuller Modular Groups" there is an example but I don't get it because I don't know much ...
1
vote
0answers
273 views

Galois group of the field of all constructible complex numbers

I am trying to understand infinite galois theory by self-made examples. First example I am struggling with is the field of all constructible (by compass and straightedge) complex numbers - let's call ...
1
vote
0answers
135 views

property of a locally cyclic group

How to prove a locally cyclic group $(G,.)$ is isomorphic to a subgroup of quotient of $\mathbb{Q}$ I've seen this statement written everywhere but without proof and I couldn't prove it (actually its ...
1
vote
0answers
44 views

Representation of a group, and finite index subspaces

Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, g\...