quotient group of cyclic group is a cyclic group.

for given cyclic group G, 

consider of canonical map f : G -> G/H

for group homomorphism g : G -> g[G] where ker g = H.

by 1st isomorphism theorem, there is an isomorphism G/H -> g[G], that is g * f^-1 

since g[G] ...

factor group이 subgroup이라는 보장이 없다.

G/H < G 라는 보장이 없다.

counter example : G=Z, H=2Z then Z/2Z = Z2 but Z2 is not a subgroup of Z

since Z has no element of order 2

then wts that quotient group of cyclic group is cyclic

for given G is cyclic and consider H which is subgroup of G

then by the thm that says subgroup of cyclic group is cyclic, H is cyclic

since G is abelian, H is normal subgroup

thus consider factor group G/H

Let G=<a> s.t |a| = n

and H = <a^m> then |a^m| = n/gcd(m,n)

consider group homomorphism f(x) = x^(n/gcd(m,n))

then ker f = H

since f(G) = <a^(n/gcd(n,m))> 

and f(G) = G/H

therefore G/H is cyclic.