Insights on coset in abstract algebra I am taking course on abstract algebra. I found the concept of coset is very important. However, I have not found any insights on my textbook (and by myself). I believe there must be great example on why coset is important in linear algebra(like solution to equations) or some other field of math. Could you provide any examples to connect coset to important theory or application?
I am NOT looking for what will appear on a standard textbook.
 A: A coset can be thought about as a copy of a group, when interpreted as a subgroup of a larger group. So anywhere that groups occur, and their subgroups are interesting at all, we expect to see cosets.
(Note that the cosets of a normal subgroup are of course much more interesting than the cosets of a generic group, because of the group structure on the set of cosets. So these arise in applications much more often.)
Geometry provides many rich examples: here are two that come to my mind immediately: An affine space is a coset, for $G$ a vector space and $H$ a subspace. The orientation of a transformation is a coset where $G$ is $O(n)$ and $H$ is $SO(n)$.
A: A subgroup $H \lhd G$ is normal iff its left and right cosets coincide, i.e. $gH=Hg$.
A: Some of the most important examples of subgroups are kernels of homomorphisms. Let $\phi: G \to H$ be a homomorphism, and $K$ its kernel. Then if two elements $g$, $h\in G$ satisfy $\phi(g) = \phi(h)$, we must have $g^{-1} h \in K$. That is, $g$ and $h$ are in the same coset of $K$. This gives a one-to-one correspondence between cosets of $K$ and things in $H$ that the homomorphism $\phi$ hits. In fact, this is the first isomorphism theorem, that $G/K \cong \text{im} \, \phi$. So we can see that cosets naturally arise "in the wild," and deserve further study.
A: Quotient groups and quotient spaces are enormously important constructions in mathematics.
