Is the Centralizer of an Element in a Free Group Nontrivial?

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In summary: It's true that every subgroup of a free group is free, but the proof is not trivial. In summary, The conversation discusses the existence of a nontrivial equation in a nonabelian free group, which would imply that certain elements commute. The centralizer of a given element in the group is found to be an infinite cyclic subgroup, which is always abelian. The fact that all subgroups of a free group are themselves free is a nontrivial result.
  • #1
Bashyboy
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Homework Statement


Note: I did not get this problem from a textbook.

Let
png.png
denote the (nonabelian) free group on the
png.png
generators
png.latex?u_1,...png
, and let
png.png
be arbitrary. My question is, does there exist a
png.png
such that
png.png
, besides
png.png
(the identity); is such an equation in the free group possible? Obviously this equation would imply they commutate, but that won't necessarily be a contradiction. It appears that I am effectively asking whether the centralizer of a given element in
png.png
is nontrivial. I can't determine the answer to this question; perhaps someone would be so kind as to guide me towards it

Homework Equations

The Attempt at a Solution

 
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  • #2
There are infinitely many other elements being ##h^k## for any integer ##k##, as ##h^{k}(h)h^{-k}=h##.
 
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  • #3
Further I am fairly confident I can prove that the set of elements that commute with ##h## is:

$$\{g^k\ :\ g\in G\wedge k\in\mathbb Z\wedge \exists m\in\mathbb N(h=g^m)\}$$
That is, powers of any element of which ##h## is a power.

But I shan't bother to write an indication of how the proof would go unless evidence appears against the fact that this thread seems to have been abandoned by the OP (or unless somebody else shows interest).
 
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  • #4
In other words, you claim that the centralizer of ##h## is ##\{g^k\ :\ g \in G,~ k \in \mathbb{Z},~\exists m \in \mathbb{N} (h=g^m)\}##?

I have another question, which seems to be related to your second post, andrewkirk. According to what I have been reading, the centralizer of a nonidentity ##h## in the free group is an infinite cyclic, i.e., a subgroup generated by one element; and evidently all subgroups of the free group are themselves free groups (wiki). Would that not make the centralizer ##C_{F_n}(h)## a free group on one generator, and therefore an abelian group, as the free group on one generator is always abelian? If so, that is rather interesting that ##C_{F_n}(h)## is always an abelian subgroup for all nonidentity ##h \in F_n##.
 
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  • #5
Bashyboy said:
Would that not make the centralizer ##C_{F_n}(h)## a free group on one generator, and therefore an abelian group, as the free group on one generator is always abelian?
Yes, it would make the centraliser a free group on one generator.
 
  • #6
Bashyboy said:
and evidently all subgroups of the free group are themselves free groups (wiki).

This is not evident at all! It's a pretty deep result.
 

What is conjugation in the Free Group?

Conjugation in the Free Group is a mathematical operation that involves changing the order of elements within a group. It is similar to the concept of conjugation in language, where the ending of a word changes based on its subject or tense.

Why is conjugation important in the Free Group?

Conjugation is important in the Free Group because it allows us to understand the structure and behavior of elements within the group. It also helps us to identify certain properties, such as commutativity, within the group.

How is conjugation performed in the Free Group?

In the Free Group, conjugation is performed by taking an element and multiplying it on the left and right by another element within the group. This results in a new element that is conjugate to the original one.

What is the difference between inner and outer conjugation in the Free Group?

Inner conjugation in the Free Group involves multiplying an element by a fixed element within the group, while outer conjugation involves multiplying an element by a varying element within the group.

How does conjugation relate to other mathematical concepts?

Conjugation in the Free Group has connections to other mathematical concepts, such as automorphisms and isomorphisms. It also has applications in group theory, topology, and algebraic geometry.

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