Give an example of a group G and a subset H s.t. H is closed under mult. but H is not a subgroup of G?

Question

abstract algebra

Answer 1

Buggi has a good answer, phrased in terms of addition. A multiplicative example would be:

G={2^n: n an integer}
H={2^n: n a positive integer}.

That's basically 2^(Buggy's answer).

Answer 2

Let G be the set of all integers, and let H be the set of all positive integers. Then, if we consider the operation of integer addition on this set, G forms a group. H is a subset of G that is closed under this operation, but it does not form a subgroup, since it does not contain additive inverses.