notes/maths/groups.md

title	tags
groups	maths, groups

Quotient groups

Wiki: https://en.wikipedia.org/wiki/Coset

Normal subgroup of G is any subgroup of G which is closed with respect to conjugates.

Theorem 1

If H is a normal subgroup of G, then aH = Ha for every a \in G.

There is no distinction between left and right cosets for a normal subgroup.

Theorem 2

Let H be a normal subgroup of G. If Ha = Hc and Hb = Hd, then H(ab) = H(cd).

Theorem 3

G/H with coset multiplication is a group.

Theorem 4

G/H is a homomorphic image of G.

Theorem 5

Let G be a group and H a subgroup of G. Then

• Ha = Hb iff ab^{-1} \in H and
• Ha = H iff a \in H

The motive for the quotient group construction is that it gives a way of actually producing all the homomorphic images of any group G.

For quotient group construction in practical instances, H is often chosen to "factor out" unwanted properties of G, and preserve in G/H only "desirable" traits.