Prove a subgroup is normal if and only if it is the union of conjugacy classes

Let G be a group and let H be a subgroup of G. Prove that H is normal if and only if it is the union of conjugacy classes.

prove properties of a ring with additive identity 0.

Let R be a ring with additive identity 0. Prove the following: (a) For all a in R, a(0) = 0. (b) a(-b)=-(ab). NOTE: see attached word document for clearer notations.

Prove the equivalence of the standard definition of equivalence relation on a group and the given alternative definition (of equivalence relation on a group).

Show that the following are equivalent: (a) ~ is an equivalence relation on a group G (b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.

Show that a+b is a unit of a commutative ring with identity

Let R be a commutative ring with identity. Let a,b in R. Suppose that a is a unit, and b^2 = 0. Show that a+b is a unit. See attached file for full problem description.

Properties of Elements of a Ring

Give an example of two elements a,b in a ring R such that a(b)=0 but b(a) <> 0. See attached file for full problem description.