Exercise 3.1.43 (Group operation on a partition)

Question

Assume ${\cal P} = \{A_i \mid i \in I\}$ is any partition of $G$ with the property that $\cal P$ is a group under the ``quotient operation'' defined as follows: to compute the product of $A_i$ with $A_j$ take any element $a_i$ of $A_i$ and any element $a_j$ of $A_j$ and let $A_iA_j$ be the element of $\cal P$ containing $a_ia_j$ (this operation is assumed to be well defined). Prove that the element of $\cal P$ that contains the identity of $G$ is a normal subgroup of $G$ and the elements of $\cal P$ are the cosets of this subgroup (so $\cal P$ is just a quotient group of $G$ in the usual sense).

Richard Ganaye · Accepted Answer

\begin{proof} 

Consider the relation defined on $G$ by 
$$a \sim b \iff \exists A \in {\cal P},\ a \in A 	ext{ and } b \in A.$$

Then $\sim$ is an equivalence relation:
\begin{enumerate}
\item[R.] Since $\cal P$ is a partition, there is some $A \in {\cal P}$ such that $a \in A$, thus  $a \sim a$.

\item[S.] If $a \sim b$ then there is some $A \in {\cal P}$ such that $a \in A$ and $b \in A$. Then $b \in A$ and $a \in A$, so $b \sim a$.   

\item[T.] If $a \sim b$ and $b \sim c$, there exists $A \in {\cal P}$ such that $a \in A$ and $b \in A$, and similarly there exists $B \in {\cal P}$ such that $b \in B$ and $c \in B$. Since $b \in B \cap C$, $B \cap C 
e \varnothing$, so by definition of a partition $B = C$. Therefore $a \in B$ and $c \in B$, so $a \sim c$.
\end{enumerate}
Note that the class $\overline{a}$ of $a$ for this relation is the unique $A \in {\cal P}$ such that $a \in A$. Indeed,
$$b \in \overline{a} \iff a \sim b \iff \exists C \in {\cal P}, \ a \in C 	ext{ and } b \in C.$$
Since $a \in A$ and $a \in C$, then $C \cap A 
e \varnothing$, thus  $C = A$, so $b \in A$. Conversely, if $b \in A$, since $a \in A$, then $b\sim a $ so $b \in \overline{a}$. This shows that $\overline{a} = A$.

For all $a \in G$ and $A \in {\cal P}$,
\begin{align}
a \in A \Rightarrow \overline{a} = A.
\end{align}
So the  associate partition relative to the relation $\sim$ is the partition $\cal P$ itself.

\bigskip
We say that the relation $\sim$ is compatible with the product in $G$ if 
\begin{align}
(a \sim a' 	ext{ and } b \sim b') \Rightarrow ab \sim a'b'.
\end{align}
In particular, since $b \sim b$,
 \begin{align}
 a \sim a' \Rightarrow (ab\sim a' b \quad 	ext{and}\quad  b a \sim ba').
 \end{align}

By hypothesis, the product in $\cal P$ is well defined, so that the class $AB$ doesn't depend of the choice of $a \in A$ or $b \in B$. Therefore the compatibility properties (2) and (3) are true.

\bigskip
Let $N$ be the unique $N \in {\cal P}$ such that $1_G \in N$ (so $N = \overline{1_G}$ by (1)). We prove that $N \unlhd G$. Let $a \in N$ and $g \in G$. Then $a \sim 1_G$, thus by (3), $ag^{-1} \sim g^{-1}$ and $g a g^{-1} \sim gg^{-1} = 1_G$, therefore $ga g^{-1} \in N$. This shows that $N \unlhd G$.

\bigskip
Let $A \in {\cal P}$. Then $A 
e \varnothing$ by definition of a partition. Let $a$ be some fixed element in $A$. By (1), every element $b \in A  = \overline{a}$ satisfies $b \sim a$. By (3), $ba^{-1} \sim aa^{-1} = 1$, thus $ba^{-1} \in N$ and $b \in aN$. Conversely if $b \in aN$, then $ba^{-1} \in N$, so $ba^{-1} \sim 1$ and by (3),  $b \sim a$, thus $b \in \overline{A} = A$. This shows that $A = aN$ is a left coset relative to $N$.

Conversely, consider a left coset $a N$, where $a \in G$. By the direct part, if $A \in {\cal P}$ is the unique element of $\cal P$ which contains $a$, then $A = aN$, so every left coset is an element of $\cal P$.

We have proved that the elements of $\cal P$ are the cosets relatives to $N$.

\bigskip
Finally, let $A, B$ be any elements of $\cal P$, and suppose that$a \in A$, $b \in B$.  In $G/N$, $aN b N = abN = cN$, where $C = cN \in {\cal P}$ satisfies $ab \in C$, so by definition the product $AB$ in $\cal P$ is equal to $C$ (so $\cal P$ is just a quotient group of $G$ in the usual sense).


\end{proof}

Exercise 3.1.43 (Group operation on a partition)

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Exercise 3.1.43 (Group operation on a partition)

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