Some results about Group Rings/Algebras

Question

I'm trying to build up my intuition on group algebras, $k[G]$ where $k$ is a field. Here are some things I'd like to know about:

1. If $H \leq G$ then is $k[H]$ a subalgebra of $k[G]$?
2. If $G_1, G_2$ are groups, what can we say about $k[G_1 \oplus G_2]$?

I don't know if 1 is true but it seems reasonable.

EDIT: My idea for this: Use the inclusion of H into G to give an inclusion of $k[H]$ into $k[G]$, which commutes with the structure map of the $\mathbb{R}$-algebras.

Also if anyone can recommend a good text discussing group rings in detail, I would be very interested.

EDIT 2: What happens if the $k$ is replaced by a general ring?

Answer 1

1. Yes, $K[H]$ is embedded naturally in $K[G]$ because $K[H]=\{a\in K[G]\mid supp(a)\subseteq H\}$. Here $supp(a)$ is the support of $a$.

2. Consider the tensor product: Lemma $3.4$ here.

3. References on text books: see here.