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Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are isomorphic. In particular, there is a nonzero endomorphism of $M$ sending $M$ onto the socle $\operatorname{soc}(M)$.

I cannot see the connection here. How does being symmetric implies composition factors? Any help would be appreciated!

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    $\begingroup$ Sorry, what's $k$, what's $N$? $\endgroup$
    – YCor
    Oct 13 '20 at 13:54
  • $\begingroup$ hi! $k$ is a field and $N$ is a group. $\endgroup$
    – user666
    Oct 13 '20 at 15:53
  • $\begingroup$ You probably also implicitly assumed that $N$ is finite. $\endgroup$
    – YCor
    Oct 13 '20 at 16:23
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For every Frobenius algebra $A$ there is a bijection $\pi$ such that $top(P_i) \cong soc(P_{\pi (i)})$ when $P_i$ denote the indecomposable projective $A$-modules. Being symmetric implies that $A$ is weakly symmetric (meaning that $\pi$ is the identity). Thus top and socle of every $P_i$ coincide which is what you asked for when I understand your question correct. For proofs and more on this see the book "Frobenius algebras I" in chapter IV. by Skowronski and Yamagata. When M is an indecomposable projective $A$-module, let $S:=soc(M)$ be the socle of $M$. Then we have a surjective map $M \rightarrow S$ (since top and socle of $M$ coincide) that induces an isomorphism $top(M)=M/rad(M) \rightarrow S$. Thus we have a surjective map $M \rightarrow soc(M)$.

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  • $\begingroup$ thank you for your input! question: what is $top(Pi)$? pardon please $\endgroup$
    – user666
    Oct 13 '20 at 12:04
  • $\begingroup$ @steven It is the top composition factor, namely $P_i /rad(P_i)$. $\endgroup$
    – Mare
    Oct 13 '20 at 12:06
  • $\begingroup$ clear now thank you! $\endgroup$
    – user666
    Oct 13 '20 at 12:12
  • $\begingroup$ hang on sir. Do we need "the two coincide" to conclude that there is a onto map from $M$ to the socle? what is the connection here? thx! $\endgroup$
    – user666
    Oct 13 '20 at 12:16
  • $\begingroup$ helpful! thx you! $\endgroup$
    – user666
    Oct 13 '20 at 12:31

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