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The real compact supergroup $S^{1|1}$ is analized from different perspectivesand its representation theory is studied. We prove it is the only (up toisomorphism) supergroup, which is a real form of $({\mathbf C}^{1|1})^\times$with reduced Lie group $S^1$, and a link with SUSY structures on ${\mathbfC}^{1|1}$ is established. We describe a large family of complex semisimplerepresentations of $S^{1|1}$ and we show that any $S^{1|1}$-representationwhose weights are all nonzero is a direct sum of members of our family. We alsocompute the matrix elements of the members of this family and we give a proofof the Peter-Weyl theorem for $S^{1|1}$.

SUSY structures, representations and Peter-Weyl theorem for $S^{1|1}$