7.3, due March 11

• Difficult: I didn't understand if a subgroup had to contain elements of the group G, or if it just had to be a group under the operation in G. Because if the subgroup H doesn't have to contain elements from G, then it doesn't seem like a subgroup to me, it seems more like a... co-group. What exactly makes group H "sub"? I think that 7.11 was saying it has to be a subset of the group G, but that's not what the definition said. So I'm confused.
• Reflective: If a center is every element in a such that ag=ga, then does that mean that every element is a center in an abelian group? That'd be nifty.