Thread: Order of elements in a group

This probably just a simple manipulation problem but I keep going in circles. It's from Introduction to Algebra by Cameron
A group which contains elements a,b,c,d,e (none equal to the identity) such that ab=c, bc=d, cd=e, de=a, ea=b. Find the orders of a,b,c,d,e.
I found the 'squares' and 'cubes' of the elements, but it starts to get ugly with higher powers. I know I'm probably suppose to be looking for when a pattern repeats for the powers, but I always end up with a horrible expression or back where I started. Any help pointing me in the right direction, please?

Originally Posted by bleys

This probably just a simple manipulation problem but I keep going in circles. It's from Introduction to Algebra by Cameron
A group which contains elements a,b,c,d,e (none equal to the identity) such that ab=c, bc=d, cd=e, de=a, ea=b. Find the orders of a,b,c,d,e.
I found the 'squares' and 'cubes' of the elements, but it starts to get ugly with higher powers. I know I'm probably suppose to be looking for when a pattern repeats for the powers, but I always end up with a horrible expression or back where I started. Any help pointing me in the right direction, please?

Interesting problem! I started by expressing everything in terms of a and b:

,
,
,
,
.

From the last two of those relations it follows that

After a good deal of experimentation I came across this:

. Multiplying on the right by , you see that , the identity element of the group. (I have to use 1 for the identity element, because e already represents a group element.)

Therefore, writing the elements in terms of a and b, (from (*)). Thus and in particular b commutes with a. You can then easily check that , and the other elements b, c, d, e also have order 11.

Thank you very much for replying! I also found out but would have never even guessed that second expression for b. Thank you for your help!