Find a subgroup of Z12 Z4 Z15 that has order 9SolutionConsid

Find a subgroup of Z_12 Z_4 Z_15 that has order 9

Solution

Consider 4 Z12 and 5 Z15.

Then <4> is a cyclic subgroup of order 3 and <5>is a cyclic subgroup of order 3.

Consider <4> <0> <5i> Z12 Z4 Z15.

Then <4> <0> <5> Z3 <0> Z3 Z3 Z3.

Therefore <4> <0> <5>is a subgroup of order 9. For three positive integers m, n, k, if lcm(m, n, k) = 9, then all m, n, k are one of 1 ,3, and 9, and at least one of m, n, k is 9.

Because 9 is not a divisor of 12, 4, nor 15, there is no element of order 9 in Z12 Z4 Z15.

Therefore there is no cyclic subgroup of order 9 in Z12 Z4 Z15.