Question

find all normal subgroups of {Z6, +}

Answer 1

any subgroup of
\[\{\mathbb Z_6,+\}\] will be normal because it is abelian.

Answer 2

normal subgroups will be the trivial subgroup
\[\{0\}\]
the subgroup of order 2 isomorphic to
\[\mathbb Z_2\]
\[\{0,3\}\]
and the subgroup of order 3 isomorphic to
\[\mathbb Z_3\]
\[\{0,2,4\}\] and the whole group

Answer 3

alright, good to know, abelian groups always have normal subgroups. thanks

Answer 4

that is any subgroup of an abelian group is normal. i am not sure what definition of normal you are using, but if it is
\[x^{-1}hx\in H \forall \text{ }h\in H \] then if it is abelian you see that
\[x^{-1}hx=x^{-1}xh=eh=h\] so all subgroups must be normal