• ABSTRACT ALGEBRA Exercise 4.1.1 Prove Proposition 4.1.5: Suppose GX ? X is an action. Then for all x ? X, stabG(x) is a subgroup of G and the kernel K of the action is the intersection of all stabilizers K = ? stabG(x). x?X The orbits OG(X) form a partition of X, and the action of G on X is transitive if and only if G x = X for some x ? X

Exercise 4.1.1 Prove Proposition 4.1.5: Suppose G X X ) X is an action. Then for all I E X, stabg{:c) is a subgroup of G and the kernel K of the action is the intersection of all stabilizers K = stabdx). IEX The orbits (90(X} form a partition of X, and the action of G on X is transitive if and only if G - a: : X for some :I: E X