Why is it interesting?

Quotient groups are crucial to understand, for example, symmetry breaking. When a group $G$ breaks to a subgroup $H$ the resulting Goldstone bosons live in the quotient space: $G/H$.

Moreover, quotient groups are a powerful way to understand geometry. Instead of a long list of axioms one can study geometry by treating the corresponding space as a homogeneous space (= coset space) and then study invariants of transformation groups of this homogeneous space.

In other words, this point of view allows us to study geometry by using the tools of group theory. This idea is known as Klein Geometry.

In addition, multi-Monopole dynamics was the first example where geodesics on a moduli space were used. (Source: page 314 in Topological Solitons by Manton, Sutcliff)

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A helpful (but slightly wrong) way to think about $G/N$ is that is consist of all elements in $G%$ that are not elements of $N$. A subgroup $N$ is defined through some special condition that its members must fulfil. Formulated differently: elements of $N$ are elements of $G$ that have some special property.

For example, the subgroup $n\mathbb{Z}$ of $\mathbb{Z}$ consist of all integers that are a multiple of $n$. (More explicitly: members of $3\mathbb{Z}\subset\mathbb{Z}$ are all integers that are divisible by $3$). Or another example: the subgroup $SO(N)$ of $O(N)$ consists of all $N \times N$ matrices with determinant equal to $1$. Now, $G/N$ consist of all elements that do not have this extra property. This isn't really correct.

We have, $N\in G/N$, but the important thing is that $N$ simply becomes the identity element of $G/N$, i.e. is trivial. This is how $N$ gets "modded out" from $G$ through the map $\varphi:G\to G/N$, which is called natural projection homomorphism. The elements are still there in $G/N$, but are mapped to the identity element. As defined above: $G/N$ is the set of all cosets and $N$ is the trivial coset. (See the examples below how this happens).

The assumption that the action of $G$ is transitive means that the structure of $M_0$ is determined by $G$ and $H=H_{\Phi_0}$ for any given $\Phi_0 \in M_0$. In fact: $$ M_0=G/H \tag{(5.37)}$$the space of right cosets of H in G. ($g_1$, $g_2$ $\in G$ are said to be in the same right coset of H in G if and only if there exists an $h \in H$ such that $g_1=g_2 h$. This defines an equivalence relation on G and the equivalence classes are the right cosets.)

Magnetic monopoles in gauge field theories by P GODDARD and DI OLIVE

Definition:

For a group $G$ and a normal subgroup of it $N$, we call

$$ G/N=\{gN:g\in G\} $$ the set of all cosets of $N$ in $G$ or equivalently the quotient group of $N$ in $G$. Thus the quotient group of $N$ in $G$ is defined as the set of all cosets of $N$ in $G$. ($G/N$ is spoken as "$G$ mod $N$").

• A great discussion of coset spaces and manifolds vs. coset spaces can be found in "Geometry of Physics" by Frankel page 456 and page 457.
• http://www.math3ma.com/mathema/2016/10/17/whats-a-quotient-group-really-part-1
• http://math.stackexchange.com/a/109114/120960
• http://www1.spms.ntu.edu.sg/~frederique/lecture9ws.pdf
• http://kias.dyndns.org/topogeo.html

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Examples

Example1

Example2:

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