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? 



Projective modules and tensor products
He claims that by the proof of Lemma 5 (which is not in google preview) it follows that $V\_2\otimes V\_p$ has a submodule isomorphic to $V\_{p-1}$, but this is the socle of $P\_{p-1}$ since $P\_{p-1}$ is the projective cover of $V\_{p-1}$ and a group algebra is always symmetric.


For the next question he first claims that we already know that $V\_2\otimes P\_{p-1}$ is projective. But then the universal property of projective shows, that $P\_{p-2}$ is a direct summand (using the homomorphic image and the the projection onto the top as the maps). Since $V\_p$ is projective, a direct application of the universal property leads the claim.


