# The number of elements not in conjugate of a subgroup

There is a proposition about the number of elements not lying in any conjugate of a subgroup when the group is finite. It states:

If $$G$$ is a finite group and $$H$$ is a proper subgroup, then the number of elements not in any conjugate of $$H$$ is at least $$|H|$$.

(See, for example, Isaac's Finite Group Theory Page 7.

However, I wonder whether the statement still holds when $$G$$ is an infinite group. I suppose it's not so hard to prove the statement for the infinite case, but I failed to find a way. Though it's clear that the statement holds when $$G$$ has an element with infinite order, I couldn’t find a way if I already get $$G$$ has a finite subgroup contains $$H$$ and the existence of a such subgroup is something I can't work out as the bounded Burnside problem shows there exists finitely generated infinite group with every element having a bounded order.

• A proof for the finite case can also be found in Proof for a subgroup $H$ of the finite subgroup $G$ there are at least $|H|$ elements that are not in conjugates of $H$. Commented Mar 16, 2019 at 20:54
• You are incorrect in claiming that the statement holds when $G$ has an element of infinite order; as noted in the answer, you can start with a torsionfree infinite group (lots of elements of infinite order), and end up with a torsionfree group containing your original group, in which any two non-identity elements are conjugate. In particular, all elements lie in a conjugate of any nontrivial subgroup $\langle x\rangle$. Commented Mar 16, 2019 at 22:17

For an infinite group, it is in fact possible that every element is contained in some conjugate of a proper subgroup containing just two elements. That is, it is possible for an infinite group to consist of just two conjugacy classes: the identity and everything else.

The construction of such a group is not elementary. A very nice outline of the process has been given by in Arturo Magidin's answer to Infinite group with only two conjugacy classes.

• Note that this example is countable. Then by a pullback argument, one can get an example of $G$ and $H$ in which $G$ is free of countable rank. Since Osin produced a (considerably harder) example for which $G$ is infinite finitely generated (with 2 conjugacy classes), one even has an example of $(G,H)$ with $G$ free of finite rank.
– YCor
Commented Mar 17, 2019 at 0:36

Here's an explicit example:

let $$G$$ be the group of increasing self-homeomorphisms of $$\mathbf{R}$$ with bounded support (i.e., identity outside a compact subset), and $$H$$ the subgroup of those increasing self-homeomorphisms with support in $$[-1,1]$$.

Indeed, if $$g\in G$$, there exists $$u\in G$$ such that $$u(\mathrm{Supp}(g))\subset [-1,1]$$, and hence $$ugu^{-1}\in H$$.

Also, if we restrict to those elements in $$G$$ (and in $$H$$) that are piecewise affine, for which slopes are integral powers of $$2$$ and breakpoints are dyadic numbers, we get a countable example for $$G$$.

• Thanks！By the way, may I ask how to construct a counterexample like that? (I know that question is a little bit too broad, but I'm just too curiuos about how to get such a counterexample, for instance, the specific one presented above.) Commented Mar 17, 2019 at 15:47
• I don't know, I don't have a particular recipe...!
– YCor
Commented Mar 18, 2019 at 19:33
The text passage discusses a proposition about the number of elements not lying in any conjugate of a subgroup in finite and infinite groups. It includes a question about whether the statement holds for infinite groups and provides a counterexample showing that it does not.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# The number of elements not in conjugate of a subgroup

There is a proposition about the number of elements not lying in any conjugate of a subgroup when the group is finite. It states:

If $$G$$ is a finite group and $$H$$ is a proper subgroup, then the number of elements not in any conjugate of $$H$$ is at least $$|H|$$.

(See, for example, Isaac's Finite Group Theory Page 7.

However, I wonder whether the statement still holds when $$G$$ is an infinite group. I suppose it's not so hard to prove the statement for the infinite case, but I failed to find a way. Though it's clear that the statement holds when $$G$$ has an element with infinite order, I couldn’t find a way if I already get $$G$$ has a finite subgroup contains $$H$$ and the existence of a such subgroup is something I can't work out as the bounded Burnside problem shows there exists finitely generated infinite group with every element having a bounded order.

• A proof for the finite case can also be found in Proof for a subgroup $H$ of the finite subgroup $G$ there are at least $|H|$ elements that are not in conjugates of $H$. Commented Mar 16, 2019 at 20:54
• You are incorrect in claiming that the statement holds when $G$ has an element of infinite order; as noted in the answer, you can start with a torsionfree infinite group (lots of elements of infinite order), and end up with a torsionfree group containing your original group, in which any two non-identity elements are conjugate. In particular, all elements lie in a conjugate of any nontrivial subgroup $\langle x\rangle$. Commented Mar 16, 2019 at 22:17

For an infinite group, it is in fact possible that every element is contained in some conjugate of a proper subgroup containing just two elements. That is, it is possible for an infinite group to consist of just two conjugacy classes: the identity and everything else.

The construction of such a group is not elementary. A very nice outline of the process has been given by in Arturo Magidin's answer to Infinite group with only two conjugacy classes.

• Note that this example is countable. Then by a pullback argument, one can get an example of $G$ and $H$ in which $G$ is free of countable rank. Since Osin produced a (considerably harder) example for which $G$ is infinite finitely generated (with 2 conjugacy classes), one even has an example of $(G,H)$ with $G$ free of finite rank.
– YCor
Commented Mar 17, 2019 at 0:36

Here's an explicit example:

let $$G$$ be the group of increasing self-homeomorphisms of $$\mathbf{R}$$ with bounded support (i.e., identity outside a compact subset), and $$H$$ the subgroup of those increasing self-homeomorphisms with support in $$[-1,1]$$.

Indeed, if $$g\in G$$, there exists $$u\in G$$ such that $$u(\mathrm{Supp}(g))\subset [-1,1]$$, and hence $$ugu^{-1}\in H$$.

Also, if we restrict to those elements in $$G$$ (and in $$H$$) that are piecewise affine, for which slopes are integral powers of $$2$$ and breakpoints are dyadic numbers, we get a countable example for $$G$$.

• Thanks！By the way, may I ask how to construct a counterexample like that? (I know that question is a little bit too broad, but I'm just too curiuos about how to get such a counterexample, for instance, the specific one presented above.) Commented Mar 17, 2019 at 15:47
• I don't know, I don't have a particular recipe...!
– YCor
Commented Mar 18, 2019 at 19:33
user
The text passage discusses a proposition about the number of elements not lying in any conjugate of a subgroup in finite and infinite groups. It includes a question about whether the statement holds for infinite groups and provides a counterexample showing that it does not.

TYPE: SINGLE_QUESTION_WITH_ANSWER