I'm having trouble understanding the motivation behind a coset. The book I'm using (A Book of Abstract Algebra) states:
Let G be a group, and H a subgroup of G. For any element a in G, the symbol
aH
denotes the set of all products ah, as a remains fixed and h ranges over H. aH is called the left coset of H in G.
It goes on to say the same about right cosets. I understand this definition (or I think I do), but what does this accomplish? Is it saying that given a subgroup H, you can generate a group G using cosets?
Thank you in advance.
Best Answer
Let me begin with a geometric example:
$\Bbb C^*$ be the group of non-zero complex numbers under multiplication
Let $H=\{x+iy\in \Bbb C:x^2+y^2=1\}$ which basically contains all the complex numbers lying on the unit circle.
Now consider the left coset $(3+4i)H$, Geometrically speaking this coset contains all points lying on the circle centered at the origin and radius $5$ (Why?)
In general the left coset $(a+ib)H$ contains all points on the circle centered at the origin and radius $\sqrt{a^2+b^2}$
Here the cosets of $H$ partition the punctured complex plane (i.e. $\Bbb C^*$) into concentric circles. Now isn't that amazing?
As seen in the above example:=
The essence of cosets lies in the fact that they partition the entire group into equivalence classes.
Moreover if the group is finite, each of the partitions will have the same number of elements, this is because of the way we define Right/Left cosets.
All the above observations leads to one of the most important result in Group Theory - The Lagrange's Theorem.