i < m , 0 j < n ) are all distinct.Let a and b be elements of a group G. Let ord(a) = m and ord( b ) = n ; let lcm(m,n) denote the least common multiple of m and n. Prove parts 1 5 :
1. If a and b commute , then ord(ab) is a devisor of lcm( m,n ).
2. If m and n relatively prime , then no power of a can be equal to any power of b (except for e). ( Remark : Two integer are said to be relatively prime if they have no common factors except ? 1.)
3. If m and n
are relatively prime , then the products aibi ( 0 i < m , 0 j < n ) are all distinct.
4. Let a and b commute. If m and n are relatively prime , then ord(ab) = mn.
5. Let a and b commute. There is an element c in G whose order is lcm[m.n].6. Give an example to show that part 1 is not true if a and do not commute.
Thus , there is no simple relationship between ord(ab) , ord(b) if a and b fail to commute.