(a) the order of a subgroup must _____ the order of the group
(b) 5 (mod 70) is an element. can yuo think of others? what is the general pattern?
(c) use the fact that φ is a homomorphism.
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.