Factor group

Dependencies:

1. Subgroup
2. Product of normal cosets is well-defined

Let $N$ be a normal subgroup of $G$. Then $G/N = \{gN: g \in G\}$, the set of all cosets of $N$, is a group under the coset multiplication operation.

Proof

• Closure: Multiplication of two cosets is a coset.
• Associativity: $(aN)((bN)(cN)) = (aN)(bcN) = (abc)N = (abN)(cN) = ((aN)(bN))(cN)$.
• Identity: $N$ is the identity, since $N(gN) = (eN)(gN) = (eg)N = gN$.
• Inverse: $(gN)^{-1} = g^{-1}N$, since $(gN)(g^{-1}N) = (gg^{-1})N = N$ and $(g^{-1}N)(gN) = (g^{-1}g)N = N$.

Dependency for:

1. Third isomorphism theorem
2. Second isomorphism theorem
3. First isomorphism theorem
4. Correspondence theorem
5. Normal correspondence theorem
6. Quotient Ring

Info:

• Depth: 4
• Number of transitive dependencies: 6

Transitive dependencies:

1. Group
2. Coset
3. For subset H of a group, a(bH) = (ab)H and (aH)b = a(Hb)
4. Subgroup
5. Normal Subgroup
6. Product of normal cosets is well-defined