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Our initial aim was to answer the question: does the Frobenius (symmetric)property transfers from a strongly graded algebra to its homogeneous componentof trivial degree? Related to it, we investigate invertible bimodules and thePicard group of a finite dimensional quasi-Frobenius algebra $R$. We computethe Picard group, the automorphism group and the group of outer automorphismsof a $9$-dimensional quasi-Frobenius algebra which is not Frobenius,constructed by Nakayama. Using these results and a semitrivial extensionconstruction, we give an example of a symmetric strongly graded algebra whosetrivial homogeneous component is not even Frobenius. We investigateassociativity of isomorphisms $R^*\ot_RR^*\simeq R$ for quasi-Frobeniusalgebras $R$, and we determine the order of the class of the invertiblebimodule $H^*$ in the Picard group of a finite dimensional Hopf algebra $H$. Asan application, we construct new examples of symmetric algebras.

Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras