Top and bottom composition factors of $M$ are isomorphic

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 $M rightarrow soc(M)$.