Let Cn be a cyclic group of order n.

A. How many sub-groups of order 4 there are in C2xC4... explain.

Put $\displaystyle C_2:=<a>\,,\,\,C_4:=<b>$, then it's easy to check that the following are sbgps. of order 4 of $\displaystyle C_2\times C_4$ : $\displaystyle \{1\}\times C_4=\{(1,b),(1,b^2),(1,b^3),(1,1)\}\,,\,\,<(a,b)>= \{(a,b),(1,b^2),(a,b^3),(1,1)\}$ , $\displaystyle <a>\times<b^2>=\{(1,1),(1,b^2),(a,1),(a,b^2)\}$ Try now to mimick the above with the group below.
B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain.

C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is its order?

If $\displaystyle H=<h>$ , then any $\displaystyle \phi \in Aut(H)$ is completely and uniquely determined by $\displaystyle \phi(h)$ , and since this must be an AUTOMORPHISM it follows that $\displaystyle \phi(h)$ has to be a generator of the group, so $\displaystyle \phi(h)=h^r$ , with $\displaystyle (r,8)=1\Longleftrightarrow\,r=1,3,5,7\Longrightarr ow |Aut(H)|=4$. Now check that this can't be a cyclic group, say by finding two different elements (i.e., automorphisms) of order 2...
D. What is the Automorphism group of an infinite cyclic group?

Just as above: the image of a generator MUST be a generator, so....what are the generators of $\displaystyle \mathbb{Z}$ and what're the possibilities of homomorphisms between them...
About A-> it's obvious (because it has index 2) that a subgroup of order 4 is normal...

Oh, dear: much easier! EVERY sbgp. of an abelian group is trivially normal... Tonio
But I can't figure out how many subgroups of this form there are...

About the other parts-I've no idea...

I'll be delighted to get guidance about all the parts in this question.

Tnx in advance...