a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then
.
b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).
c) Give an example of an Abelian group G and elements a and b in G such that o(ab)
o(a)o(b). Compare part (b)
I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then
iff n is a divisor of t)
But, I just don't know what to do with part a...
From what I understand,
and
I have done all sorts of random inserts of
,
, etc. Nothing seems to give me what I need though... erm, I was wondering, since
does
?