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.
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