My question(s) relate(s) to pp51-52 of Local Representation Theory by JL Alperin -- the relevant pages are contained in the Google Books preview http://books.google.com/books?id=p7ylsZUmK3MC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false. In these pages he is dealing with representations of $SL(2,p)$ over a field $k$ of characteristic $p>0$. He has previously constructed the $p$ simple modules $V_1,\ldots,V_p$, and wants now to describe composition series for the corresponding projective indecomposable module $P_i$ when for $i$ less than $p$ (he deals with $P_p\cong V_p$ later, but assumes we know this isomorphism).

The book has a certain fluency with the notion of projective module which I haven't picked up yet. On p51, after having proved the Lemma that $V_2\otimes V_n\cong V_{n-1}\oplus V_{n+1}$ (if $n$ is less than $p$), he proceeds to show that $V_2\otimes V_p\cong P_{p-1}$.

His elegant proof just claims that $V_2\otimes V_p$ has a summand isomorphic with a submodule of the socle of $P_{p-1}$. I see how this proves what we want but I don't see how to show this simply.

On the next page, I can't even prove what he's claiming:

We examine $P_{p-2}$ by studying $V_2\otimes P_{p-1}$. Since $V_p$ is projective and since $V_{p-2}$ is a homomorphic image, we have that $V_2\otimes P_{p-1}$ has a direct summand isomorphic with $P_{p-2}\oplus V_p\oplus V_p$. Why is this?