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If a group G is non abelian and a and b in G have orders |a|= n and |b|= m, is there a correlation we can draw between m and n and |ab|?
in general, no. it may be that |ab| isn't even finite.
for example, if G is the quotient of the free group on two generators defined by:
a2 = b2 = e
then ab is not of finite order, even though both a and b are of order 2.
Thanks. That's what I thought.
So How do I go about figuring out the order of different elements in a group. For instance, consider Dihedral D9 then I can define as : {a,b| |a|=9 and |b|=2}
Now if I'm trying to figure out the different Sylow 2-subgroups, I need to get the elements with order 2. How do I do it?