Question

Suppose group contains elements a and b such that |a|=4 and |b|=2 , and a^b=ba. then |ab|=?

Answer 1

5down voteaccepted

Those are orders of elements.

The order of an element g∈Gg∈G, denoted |g||g|, is defined as the smallest positive integer nn such that gn=egn=e, the identity element of the group.

So, in your question, we know that a4=ea4=e (and no smaller power will do), and that b2=eb2=e (and no smaller power will do).

This is enough to conclude that |ab|=2|ab|=2, because (ab)2=abab=a(ba)b=a(a3b)b=a4b2=e⋅e=e(ab)2=abab=a(ba)b=a(a3b)b=a4b2=e⋅e=e, and abab cannot be the identity, because otherwise a would be the inverse of b, and so have the same order.

shareciteimprove this answeranswered Sep 16 '13 at 4:38William Ballinger2,9081022add a commentup vote1down vote

As a hint, start taking powers of abab. First note that if

ab=e⟹a=b−1⟹a2=2ab=e⟹a=b−1⟹a2=2

contradicting that |a|=2|a|=2. Next we have

(ab)2=abab=a(a3b)b=a4b2=...?