Semisimplicity for tensor products of representations of finite groups Let $G$ be a group and $k$ a field of characteristic $p>0$. Let $$\rho_i: G\to GL(V_i),~ i=1,2$$ be two finite-dimensional semisimple $k$-representations of $G$, with $\dim(V_1)+\dim(V_2)<p+2.$  Then a 1994 theorem of Serre tells us that $\rho_1\otimes\rho_2$ is semisimple.

I was wondering -- is there is an easier proof in the case that $G$ is finite?

Specializing Serre's proof to the case of finite groups seems not to yield any real simplification; one has to apply the so-called saturation technique to replace the subgroup of $G$ generated by elements of order a power of $p$ by a linear-algebraic group. 
I suspect the answer is "no" -- it seems to me that the general case reduces to the case of finite groups (by a spreading out and specialization argument), so it's hard to believe this case could be substantially easier -- but I figured it was worth asking. I'd also be interested in a proof with worse bounds, e.g. with $p+2$ replaced by any increasing function of $p$.
 A: There is a result of D. S. Passmann and D. Quinn in "Burnside's theorem for Hopf algebras", Corollary 8, which says the following:

If $A$ is a finite-dimensional Hopf algebra, then the set of semisimple
  $A$-modules is closed under tensor product if and only if the Jacobson
  radical $J(A)$ is a Hopf ideal of $A$.

If you require all semisimple $A$-modules to be closed under tensor products, then the question reduces to something about the Jacobson radical.  
In a paper of M. Lorenz, "Representations of Finite-Dimensional Hopf Algebras", he makes the following comments:

Remarks and Examples. 1) If  all  simple $H$-modules are 
  1-dimensional (equivalently, $H/J\simeq k^r$ as $k$-algebras  for  some $r$), then  all 
  tensor  products  of simple $H$-modules are  1-dimensional as  well, and
  hence  condition (2) of the lemma is clearly satisfied. Thus $J$ is a
  Hopf ideal in this case. ......
2) If $H=kG$ is a finite group algebra, then $J$ is a
  Hopf ideal precisely if $G$ has a normal Sylow $p$-subgroup  [M].

[M] R. K. Molner, "Tensor products and  semisimple modular representations  of finite groups  and  restricted  Lie  algebras", Rocky  Mountain J.  Math. 111981 ,  581-591.
