## Mathematical detour: normal subgroups

Before continuing our exploration of Pauli stabilisers, we need a bit of pure mathematics.

Let H be a subgroup of G, written H\leqslant G.
We say that H is a **normal** subgroup of G, and write H\triangleleft G, if H is invariant under conjugation by all elements of G, i.e. ghg^{-1}\in H for all g\in G and all h\in H.

Note that we only require that ghg^{-1} be some arbitrary element in H, not that ghg^{-1}=h.

If H\leqslant G is an arbitrary (not necessarily normal) subgroup, then we can use it to “slice up” G into subsets of equal size called **cosets**, one of which is H itself.
We define a (left) coset to be a set of the form
gH = \{gh \mid h\in H\}
for any fixed g\in G.
Any two cosets (i.e. any two choices of g\in G) are either entirely equal or completely disjoint.
The relevance of normality here is that if H is normal, then there is no need to distinguish between left (gH) and right (Hg) cosets, and in this case we can construct the **quotient group** G/H consisting of cosets with the operation defined by gH\cdot g'H=(gg')H.
Here is one way to visualise a partition into cosets:

The bottom row represents the subgroup H, and each row above represents a coset, i.e. a set of elements generated by picking an element c_k of G that does not belong to H nor to any of the previously generated cosets, and then multiplying this element by all elements in H, one at a time.
This picture above shows that, for any finite group G and any subgroup H\leqslant G,
|G| = |G:H|\cdot|H|
where |G:H| is the number of cosets of G given by H.
This fact is known as **Lagrange’s theorem** (although Joseph-Louis Lagrange was a rather prolific mathematician, working in many areas, so this is only one of the theorems to bear his name).

It seems like it was Évariste Galois who recognised that *normal* subgroups were worthy of special attention.
Given an arbitrary subgroup H\leqslant G, we can construct a larger subgroup K\leqslant G in which H is normal, i.e. such that H\triangleleft K\leqslant G.
The largest such subgroup K is called the **normaliser of H in G**, denoted by N_G(H), and we can construct it explicitly:
N_G(H) = \{g\in G \mid ghg^{-1}\in H\text{ for all }h\in H\}.
In words, the normaliser consists of the set of elements of G that conjugate all elements of H to elements of H.
This suggests a very subtle question: is every subgroup of a normal subgroup normal?
The answer is most definitely *no*: if H\triangleleft K and K\triangleleft G then it is *not* necessarily the case that H\triangleleft G, merely that H\leqslant G.

As we shall soon see, Pauli stabilisers are not normal subgroups of \mathcal{P}_n, and we will instead want to study their normalisers.