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I am trying to find more generalized counterparts of some well-known results from modular group representations.

My question is the following:

Suppose that $H$ is a finite $p$-group acting as automorphisms on a finite dimensional $k$-algebra $A$. Suppose moreover that the field $k$ has characteristic exactly this prime $p$. Does it follow that all simple $A\#kH$ modules are in bijection with all simple $A$-modules? Obviously, I am talking about isomorphism classes of simple modules.

In the case $A=kL$ for a $p'$-group $L$ I think this result is well known although I cannot identify a reference. At least a treatment of this case can be found in Section 8.4 of Webb's book:

http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf

In this case, there is a canonical projection $A\# kH\rightarrow kH$ which makes $kH$ a canonical $A\# kH$-module. It turns out that this is a projective module and it is the projective cover of the trivial module.

Any reference to this type of smash products is very welcomed.

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Take $p=2$, $k$ algebraically closed, $A=kC_3$, and $H=C_2$ acting non-trivially on $C_3$.

Then $A$ has three simple modules, but $A\# kH$ is the group algebra $kS_3$, which has two simple modules.

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  • $\begingroup$ Ok, thanks a lot! It was easy and this example is also in the book I mentioned. Maybe it is true that always the number of simple modules of $A\#kH$ is at most the number of simple modules of $A$. $\endgroup$ – muser17 Feb 15 '17 at 13:13

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