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 $G$. * (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.

Concerning (a), the first times you encounter it, an _a fortiori argument_ can be tricky, despite its simplicity. You know $a b = ba$ holds _for all elements $a, b \in G$_ , and the elements of $H$ are just _some of the elements of $G$_.