Q:

Two questions about isomorphisms of groups

Let $A_1$ and $A_2$ be Abelian groups and let $C$ and $B_i$ be subgroups of $A_i$ for $i=1,2$ and $B_1 \cap B_2 = A_1 \cap A_2 = \{0\}$. Is then true that
$$(A_1\bigoplus A_2) / (B_1\bigoplus B_2) \ \approx (A_1 /B_2) \bigoplus  (A_1/ B_2)$$
and
$$(A_1\bigoplus A_2) /C \approx (A_1 /C) \bigoplus  (A_2/C)\text{?}$$
I am asking this (two) question(s), because I think the answer is yes (I have written isomorphisms), but intuition tells me that I am wrong in at least one case. I went trough the process and I think both isomorphisms are well defined etc.
EDIT: the second question is almost stupid, because $A_1 \cap A_2 = \{0 \}$, therefore $C = \{ 0\}$.

A:

In the first case you've written the rule incorrectly, it should be
$$(A_1 \oplus A_2)/(B_1 \oplus B_2) \simeq (A_1/B_1) \oplus (A_2/B_2)$$
and that is indeed correct.  The isomorphism is induced from the quotient maps $A_i \to A_i/B_i$.
The second one, on the other hand, doesn't make sense.  You say $A_1 \cap A_2 = \{0\}$ but then you write $A_1/C$ and $A_2/C$ as if $C \subseteq A_1 \cap A_2$.  I guess if we take $C = 0$ then all this is consistant and the isomorphism you've written is correct (well, interpreting $A_1$ and $A_2$ as subgroups of a larger group and the direct sum to be the internal one), but I'm guessing you don't intend $C = 0$ to be the only possibility.  So on this second one I'd say you are most likely a little confused.

