Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 3 / b / Solution

Write $P\cong kGe$ for a primitive idempotent $e$. The coefficient-of-identity form makes $kG$ a symmetric algebra. Associativity of this nondegenerate form identifies the orthogonal complement of $J(kG)e$ in $kGe$ with the elements annihilated by $J(kG)$, namely $\mathrm{Soc}(P)$. It therefore induces a nondegenerate $kG$-invariant pairing betweenand $\mathrm{Soc}(P)$. Both are simple by projectivity and part (a), and the symmetric form has identity Nakayama permutation. Consequently the head and socle of an indecomposable projective group-algebra module satisfy