Mark
Posted on Sunday, 12 October, 2003 - 12:01 pm:
The multiplicative group, G has eight elements

e, a, a2, a3, b, a b, a2 b, a3 b, where e is the identity element, a4=b2=e and b a=a3 b.

(i) Find the order of a2 b and find two subgroups of G of order 4.

(ii) The group H, of order 8, has elements e[1/4]k i p, where k=0, 1, 2, ..., 7. The group operation for H is complex multiplication. Determine whether G and H are isomorphic and justify your conclusion.

My problem is how to find the order of all the elements in G with the information given. Any help will be much appreciated.

Kerwin Hui
Posted on Sunday, 12 October, 2003 - 02:55 pm:
Using a4 =e, we have a3 =a-1 and hence ba=a-1 b. Can you now see that every element of the form an b has order 2? Working out the order of elements of the form an is easy.

Kerwin
Mark
Posted on Sunday, 12 October, 2003 - 04:18 pm:
I can't see how any element of the form an b has order 2! Could you please expand on it.
Thanks
Julian Pulman
Posted on Sunday, 12 October, 2003 - 04:35 pm:
So, bab-1 = a-1
=> ban b-1 = a-n
=> an ban b-1 = e
=> an ban b = e
=> (an b)2 = e
Mark
Posted on Sunday, 12 October, 2003 - 04:51 pm:
Thank you all very much