Let ϕ: ℤ70 →ℤ5 be the group homomorphism that sends a class [a]70 to [a]5
(a) Determine the subgroup lattice of ℤ70
(b) Identify the subgroup of ℤ70 which is the kernel of ϕ
(c) Is the kernel a cyclic group?
This is what I have so far
(a) Z₇₀ has lattice corresponding to the factors of 70:
(Dropping coset notation for simplicity,)
.........Z₇₀ = <1>
......./......|......\
....<2>..<5>..<7>
..../....X......X....\
<10>..<14>..<35>
........\.....|....../
......<70> = {0}
(b) ker ϕ = {[x]₇₀ in Z₇₀ : ϕ([x]₇₀) = [x]₅ = [0]₅} = <14>.
(c) Yes; subgroups of a cyclic group are cyclic.