Product of normal cosets is well-defined

Dependencies:

  1. Normal Subgroup
  2. For subset H of a group, a(bH) = (ab)H and (aH)b = a(Hb)

Let $N$ be a normal subgroup of $G$. This means that $gN = Ng$ for all $g \in G$. The coset of a normal subgroup is called a normal coset.

Define the product of cosets $aN$ and $bN$ as \[ (aN)(bN) = (ab)N \]

This product is well-defined, which means that it does not depend on the representation of the coset. This means that \[ a_1N = a_2N \wedge b_1N = b_2N \Rightarrow (a_1N)(b_1N) = (a_2N)(b_2N) \]

Proof

\begin{align} & (a_1N)(b_1N) &= (a_1b_1)N \\ &= a_1(b_1N) &= a_1(b_2N) \\ &= a_1(Nb_2) &= (a_1N)b_2 \\ &= (a_2N)b_2 &= a_2(Nb_2) \\ &= a_2(b_2N) &= (a_2b_2)N \\ &= (a_2N)(b_2N) \end{align}

Dependency for:

  1. Second isomorphism theorem
  2. Factor group

Info:

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