How to prove that this group has at most order 16

Question

If $$G=\langle a,b:a^8=b^2a^4=ab^{-1}ab=e\rangle,$$

how can I prove that $G$ has order at most $16$?

I have played with the relations for a while, but am literally stuck. I know that the order of $b$ has to be at most equal to $4$, and I also found that $aba=b$, but I don't really know how to connect these relations in order to prove that $G$ has at most $16$ elements. I am thinking that maybe Von Dyck's theorem might be useful. I found this article http://buzzard.ups.edu/courses/2012spring/projects/clausen-groups-16-ups-434-2012.pdf and don't know which might be the group I should be looking at in order to use Von Dyck's theorem, or if there is another way to approach this problem.

Any help is much appreciated.

Answer 1

This group is the generalized quaternion group $Q_{16}$ of order $16$. Try to show that the only (pairwise distinct) elements of $G$ are:

$1, a, b, a^2, b^2, ab, a^3, ab^2, a^2b^{-1}, b^{-1}, a^{-2}, a^{-1}b, ab^{-1}, a^{-1}, a^2b, a^{-1}b^{-1}$.

Answer 2

Quick fix

Well, this is a finite group and the task is to compute it. I have received lots of downvotes in the past for similar questions, but my answer is to plug the problem into a computation system and see what happens.

Simply because it does not take much time and might point at something ugly at an early stage. This is practice.

Second is to take all relations, and manually compute the group. Relations will collapse some sections of the multiplication table. You start with 8 by 4 table (8 for a's, 4 for b's), populate it with products like in a high school math table, apply relations to collapse it.

I will show the quick and dirty way, and hope not to get a wave of hate.

This is for Magma, you open an online calculator http://magma.maths.usyd.edu.au/calc/

Paste this code

$$\text{G<a,b>, h:=Group<a,b|a^8, b^2*a^4,a*b^-1*a*b>; G; Order(G)}$$

Again, my reply is not for haters of Magma and Wolfram Mathematica, but a quick fix practical solution.

Answer 3

Cayley table

For manual work you can set up the Cayley table https://en.wikipedia.org/wiki/Cayley_table

This is a related question

Completing Cayley table for a group