I never used that theorem. I simply do it by definition: (i) H is a non-trivial subset of G (ii) if a*b are in H for all a,b in H (iii) e is in H (iv) if a in H then a^(-1) in H (v) * is associate (but that follows because G is a group).I have a problem, here is the appropriate theorem:

Theorem:Let be a group with operation , and let be a subset of . Then is a subgroup of iff

(a) is nonempty

(b) if and , then , and

(c) if , then

i've used this theorem over and over in studying, but i have trouble proving (b) and (c) in the following problem[B]

Try doing it this way.