In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
We study SUSY $N$-supergroups, $N=1,2$, their classification and explicit realization, together with their real forms. In the end, we give the supergroup of SUSY preserving automorphism of $mathbf{C}^{1|1}$ and we identify it with a subsupergroup of
the SUSY preserving automorphisms of $mathbf{P}^{1|1}$.
We study a generalization of the results in cite{cfk} to the case of $SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its finite dimensional complex irreducible representations, we give a reducibility result for representations
not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end we give the Peter-Weyl theorem for $S^{1|2}$.
The purpose of this paper is to describe the projective linear supergroup, its relation with the automorphisms of the projective superspace and to determine the supergroup of SUSY preserving automorphisms of ${mathbf P}^{1|1}$
