Multiplying cycles together

Hi, I can manage to multiply 2 things together if they are of the form say:

A=( 1 2 3 4 5 6 7 8 9 )

6 3 7 9 4 2 1 8 5

B=( 1 2 3 4 5 6 7 8 9 )

3 5 1 2 7 9 6 4 8

I.e AB = ( 1 2 3 4 5 6 7 8 9 )

7 4 6 3 1 5 2 9 8

However I'm then asked to write A and B in cycle notation so

A= (1 6 2 3 7)(4 9 5)

B= (1 3)(2 5 7 6 9 8 4)

And then asked to use this to get the answers of AB and BA in cycle form too, but I really can't figure out how this is done, any help would be appreciated!

Thanks,

Nathan

Re: Multiplying cycles together

In AB, B takes 1 to 3, and A will take 3 to 7, so 1 -> 7 (remember, A and B are essentially functions on their domains). In B, 2 goes to 5, and in A 5 goes to 4, so 2 ->4 is the new rule. In summary, 1->7, 2->4, 3 -> 6, 4 -> 3, 5 -> 1, 6 -> 5, 7 -> 2, 8 -> 9, 9 -> 8 (since 8 cycles to itself in A). Now, putting this all together, begin with any number, like 1, and follow the trail until you've returned to 1 again. (1724365)(89) is the new composed rule AB. Your turn for BA.