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We develop a theory of right group-like projections in Hopf algebras linkingthem with the theory of left coideal subalgebras with two sided counitalintegrals. Every right group-like projection is associated with a left coidealsubalgebra, maximal among the ones containing the given group-like projectionas an integral, and we show that that subalgebra is finite dimensional. Weobserve that in a semisimple Hopf algebra $H$ every left coideal subalgebra hasan integral and we prove a 1-1 correspondence between right group-likeprojections and left coideal subalgebras of $H$. We provide a number ofequivalent conditions for a right group-like projections to be left group-likeprojection and prove a 1-1 correspondence between semisimple left coidealsubalgebras preserved by the squared antipode and two sided group-likeprojections.  We also classify left coideal subalgebras in Taft Hopf algebras $H_{n^2}$over a field $\mathbb{k}$, showing that the automorphism group splits them into  - a class of cardinality $|\mathbb{k}|-1$ of semisimple ones which correspondto right group-like projections which are not two sided;  - finitely many semisimple singletons, each corresponding to two sidedgroup-like projection; the number of those singletons for $H_{n^2}$ is equal tothe number of divisors of $n$;  - finitely many singletons, each non-semisimple and admitting no rightgroup-like projection; the number of those singletons for $H_{n^2}$ is equal tothe number of divisors of $n$.  In particular we answer the question of Landstad and Van Daele showing thatthere do exist right group-like projections which are not left group-likeprojections.

Integrals in left coideal subalgebras and group-like projections