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)

Layman

Student

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

Researcher

Examples

Example1

Example2:

FAQ

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