Proving G=1: Exercise from Serre's Book Trees

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In summary, the conversation discusses an exercise from Serre's book "Trees" which involves proving that a given group G is equal to the identity element. The participants consider different approaches, such as using Todd Coxeter coset enumeration or simplifying products of generators, but are unsure how to proceed. One potential first step is to prove that the orders of the generators are finite, which is discussed in Suzuki's book with hints provided. However, in this exercise, the finiteness of G cannot be assumed beforehand.
  • #1
Ultraworld
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This is an exercise from a book from Serre called Trees. Given the group

G = < a, b, c | bab−1 = a2, cbc−1 = b2, aca−1 = c2 >

I have to prove G = 1.

I don't have a clue. Of course G' = G (commutator subgroup equals the group itself) but I don't know what to deduce from that. Another first step could be to prove that the orders of a, b and c are finite. But I do not even know how to that.

If anyone could put me in the right direction i would be very grateful.

edit: I am going to try to use Todd Coxeter coset enumeration.
 
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  • #2
I doubt wetter ToddCoxeter CE will help me :frown:
 
  • #3
Since the question uses a,b,c and it's Serre, let's assume that the fact there are 3 generators is important.

The only thing I can think of doing is considering products like abc or bac etc and simplifying in two ways until ended up with something that points towards the statement a=a^2, or similar.

(No, I\ve not solved this - I'll get a pen and paper and think about it some more later)
 
  • #4
well, if you can prove all the 3 generators are of finite order than I can finish this question. Because Suzuki has this exercise in his book where he assumes G is finite and he also gives lots of hints.

Here however I can not a priori assume G is finite.
 

1. What is the significance of proving G=1 in Serre's Book Trees?

Proving G=1 is significant because it serves as a foundational step in understanding the structure of trees and their associated groups. It also highlights the idea that trees are a special case of graphs, and their properties can be analyzed using group theory.

2. How is G=1 proven in Serre's Book Trees?

The proof of G=1 in Serre's Book Trees involves using the properties of cosets and normal subgroups to show that the only possible group structure for a tree is the trivial group, G=1. This is done by considering the action of the group on the tree and analyzing its orbits and stabilizers.

3. Why is it important to understand the group structure of trees?

Understanding the group structure of trees allows us to apply group theory to analyze and classify different types of trees. It also has applications in other fields, such as computer science and physics, where trees are used to model complex systems.

4. Can the proof of G=1 be extended to other types of graphs?

Yes, the proof of G=1 in Serre's Book Trees can be extended to other types of graphs by considering the group actions on the graph and analyzing their orbits and stabilizers. However, the proof may become more complex and may not hold for all types of graphs.

5. What are some real-world applications of the proof of G=1 in Serre's Book Trees?

The proof of G=1 has applications in various fields, such as computer science, network analysis, and physics. For example, it can be used to analyze the flow of information in a computer network or to understand the behavior of particles in a physical system.

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