How many distinct subgroups does Z/36Z have? How would I begin solving this?
things to remember with cyclic groups:
1) every subgroup is cyclic.
2) if G is cyclic of order n, and k divides n, G has a UNIQUE subgroup of order k (so it has one, and only one, for each divisor k of n).
in particular, if G = <a>, with |a| = n, and k|n, so that n = kd, then the subgroup of order k is <a^{d}>.
for ABELIAN groups, a^{d} is usually written da. since Z/36Z is generated by 1+36Z = [1], we need only consider <d[1]> = <[d]> for each divisor d of 36.
the divisors of 36 are:
1,2,3,4,6,9,12,18, and 36.
your turn.
Okay, so the divisors are 1, 2, 3, 4, 6 9, 12, 18, 36. When the order is 36, the generator is <1>. When the order is 18, the generator is <2>.
So, we have the following:
order 36, <1> = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0}
order 18, <2> = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 0}
order 12, <3> = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 0}
order 9, <4> = {4, 8, 12, 16, 20, 24, 28, 32, 0}
order 6, <6> = {6, 12, 18, 24, 30, 0}
order 4, <9> = {9, 18, 27, 0}
order 3, <12> = {12, 24, 0}
order 2, <18> = {18, 0}
order 1, <0> = {0}
Am I on the right track?