Question

Group Theory. Let a and b be elements of a group G. Let ord(a)=m and ord(b)=n. prove that if m and n are relatively prime, then no power of a can be equal to any power of b (except for e).

In doing this proof, i used the fact that m cannot divide n and n cannot divide m, hence they have no common divisors. therefore m cannot equal n. but i feel like i am missing something. Any help would be appreciated.

Answer 1

i guess you could work by contradiction and assume that \(a^k=b^j\neq e\)

Answer 2

then \((a^k)^n=(b^j)^n=(b^n)^j=e^j=e\) so the order of \(a^k\) divides \(n\)

Answer 3

similarly the order of \(b^j\) divides \(m\)

Answer 4

but since there are no common divisors except for \(1\) that means the order of \(b^j\) is one and so \(b^j=a^k=e\)

Answer 5

it is not a matter of \(n\) not being equal to \(m\)

Answer 6

is it about m and n being the least positive integers?

Answer 7

yes

Answer 8

i see. thank you for your help!!

Answer 9

yw