basic coset stuff

[math_stresser]

This is a very long problem, but I can do most of it.

Let G=[a] be a cyclic group of order 10. Let H=[a^5] be a subgroup of G generated by the element a^5 from G.
List the distinct elements of H and give the order of H.
H={e,a^5}
ord H= 2

Give the index of H in G.
5

Explain why we know H is a normal subgroup in G without explicitly checking cosets.We know H is a normal subgroup in G because G and H are cyclic. (RIGHT?!?)

List the elements of the quotient group G/H.
(e)G= {e, a^5}
(a)G= {a,a^6}
(a^2)G= {a^2, a^7}
(a^3)G= {a^3,a^8}
(a^4)G= {a^4, a^9}

Determine if G/H is a cyclic group. If not, explain why G/H is not a cyclic group. If so, give a generator of G/H.
Yes G/H is a cyclic group.
generator=(a)G

Now this is where I need some help...

Let theta: G-->G/H be the natural homomorphism map from G to G/H. Illustrate the application of the natural map on the elements a^2 and a^7 in G.
Would I just show that both e and a^5 map to (e)G, a and a^6 map to (a)G, etc?
Also does that mean that the map shows a^2 and a^7 mapping to (a^2)G?

Give the kernel of the natural matp from G to G/H.
Is the kernel just e and a^5?

Is this natural map theta: G--? G/H an isomorphism? Explain.
If I am right on the previous two questions... I would say that, no, it's not an isomorphism because it's not injective.