Obvious Group and subgroup questions. My awesome math prof posted a practice midterm but didn't post any solutions to it :s
Here is the question.
Let $G$ be a group and let $H$ be a subgroup of $G$.


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*(a) TRUE or FALSE: If $G$ is abelian, then so is $H$.

*(b) TRUE or FALSE: If $H$ is abelian, then so is $G$.


Part (a) is clearly true but I am having a bit of difficulty proving it, after fulling the conditions of being a subgroup the commutative of $G$ should imply that $ab=ba$ somehow.
Part (b) I am fairly certain this is false and I know my tiny brain should be able to find an example somewhere but it is 4 am here :)
I want to use some non-abelian group $G$ then find a generator to make a cyclic subgroup of $G$ that is abelian.
Any help would be appreciated, I have looked in my book but I can't seem to find for certain what I am looking for with what we have cover thus far.

(b) Take the group of $(n \times n)$-matrices with $\mathbb{R}$-coefficients with usual matrix multiplication as G and let H be the subgroup of diagonal matrices. H ist abelian, but G is not abelian.
