# Thread: Group & Number Theory Question

1. ## Group & Number Theory Question

From Pinter 'A Book on Abstract Algebra' 2nd Ed.

Question E-2 from Ch. 10 on page 109

Let a and b be elements of a group G. Let ord(a)=m and ord(b)=n; let lcm(m,n) denote least common multiple of m and n. Prove if m and n are relatively prime, then no power of a can be equal to any power of b (except for e(the identity)).

You need to know the order of any element in a group is the order of the subgroup <x>. So suppose $j$ and $k$ are such that $a^j=b^k$. Since $<a^j>\subseteq <a>$, Lagrange says the order of $a^j$ divides m. In the same way $<a^j>\subseteq <b>$ and so the order of $a^j$ divides n. Hence the order of $a^j$ is 1; i.e. $a^j=e$.