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Let (U(t))tÃ¢Â‰Â¥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0>0, U(t0) is a Fredholm (resp., semi-Fredholm) operator, then (U(t))tÃ¢Â‰Â¥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))tÃ¢Â‰Â¥0 can be imbedded in a C0-group on X. Also we study semigroups which are near the identity in the sense that there exists t0>0 such that U(t0)Ã¢ÂˆÂ’IÃ¢ÂˆÂˆÃ°ÂÂ’Â¥(X), where Ã°ÂÂ’Â¥(X) is an arbitrary closed two-sided ideal contained in the set of Fredholm perturbations. We close this paper by discussing the case where Ã°ÂÂ’Â¥(X) is replaced by some subsets of the set of polynomially compact perturbations

Remarks on embeddable semigroups in groups and a generalization of some Cuthbert's results