If there for a finite group exists a finite subgroup of less order does it have to have a coset?

Question

If there for a finite group G exists a subgroup H with lower order than G does H has to have a coset?

My thinking is that it has to be the case, as the elements not in H at least for one subgroup, that group having to be a coset. Am I completly off?

Answer 1

Yes, every subgroup has cosets. Will come back to this later.

In the second paragraph you say " as the elements not in H at least for one subgroup, that group having to be a coset". It sounds like you think that some elements outside of $H$ can form a subgroup. This is never true. A subgroup must by definition contain the identity $e$, so any two subgroups have at least one element in common, $e$. Thus it's impossible for a subgroup to be totally from outside another subgroup.

A coset is a subset of the group of the form $gH$, where $g$ is any element in $G$. Thus $eH = H$ is a coset, so there is at least one coset. If you another element $g$ outside of $H$, $gH$ is a coset as well.

Answer 2

A (left) coset of $H$ is precisely a set of the form $gH=\{gh|h\in H\}$ where $g\in G$, so $H=1_GH$ (where $1_G$ is the identity element of $G$) is itself a coset.

Note that there is no restriction on the order of $H$, so $G$ is also a coset (of $G$)