TITLE: If G is not commutative, then is there always a subgroup that is not a normal subgroup?
QUESTION [17 upvotes]: I was having a discussion with a friend of mine about some normal group properties and then came up with the question "if G is not commutative, then is there always a subgroup that is not a normal subgroup?" It's probably more easy to solve this in the following form:$$\forall H \leq G : H  \lhd G \Rightarrow \forall a,b \in G : ab=ba$$
My question is, can anybody give a proof, or a counter-example (because I don't think it holds) of this theorem? Thanks!

REPLY [28 votes]: The answer is no.  The Quaternion Group provides the smallest counter example.
Another way to write your question is the following: "Does there exist a non Abelian group all whose subgroups are normal."  Such counter examples to your above conjecture actually have a specific name, and can be completely classified.   These are called Hamiltonian Groups.