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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?

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    $\begingroup$ 1. Yes, if you appropriately identify elements of $k\left[H\right]$ with elements of $k\left[G\right]$. A cleaner statement would be "Any group homomorphism $H \to G$ induces a $k$-algebra homomorphism $k\left[H\right]\to k\left[G\right]$, and if the former is injective, then so is the latter.". $\endgroup$ – darij grinberg Nov 21 '16 at 22:30
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    $\begingroup$ 2. What would your guess be? (What is an operation on $k$-algebras similar to the direct sum operation on groups?) $\endgroup$ – darij grinberg Nov 21 '16 at 22:30
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    $\begingroup$ Group algebras play a great role in Peter Webb's book on representation theory ( math.umn.edu/~webb/RepBook ), although I'm not sure in how much detail he goes about their basic properties. $\endgroup$ – darij grinberg Nov 21 '16 at 22:33
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    $\begingroup$ I think the only place you'll find a discussion of group algebras is places they're used for other things. For instance, you could look at the book Representations and Characters of Groups by James & Liebeck, where they're used because they're relevant to group representations $\endgroup$ – Alex Mathers Nov 21 '16 at 22:34
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    $\begingroup$ Nothing develops confidence as actually doing things! It is infinitely more useful to ask this sort of question by writing the details of what you did. $\endgroup$ – Mariano Suárez-Álvarez Nov 21 '16 at 22:41
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  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.

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  • $\begingroup$ Thanks! I haven't learned about the tensor product yet, so I've got that to look forward to. I guess if you replace $k$ with any ring, all the results still hold (but with ring isomorphisms instead of algebra ones)? $\endgroup$ – chilliBeanDream Nov 21 '16 at 22:44

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