Understanding $(12)(23)=(132)$ in the symmetric group $S_3$.

Question

I am studying symmetric groups, as an example for the symmetric group $S_3$ I have the equality:

$$(12)(23)=(132)$$

It may be very trivial but I cannot derive it: how can this be done?

Answer 1

Given the answer $(1\ 3\ 2)$, I assume that you are meant to compose these from left to right (contrary to the more common convention).

You might find the following figure to be helpful.

If anyone is curious, here is the code I used to generate this in Tikz

\begin{tikzpicture}[mystyle/.style={circle,draw,fill=gray!40}]
%"mystyle" is a variable that specifies the node's style when used as optional arg
  \foreach \i in {1,...,3}
    \foreach \j in {1,2,3} 
      {\pgfmathtruncatemacro{\label}{(\i)}
      \node[mystyle]  (\i\j) at (6*\j,-1*\i) {\label};}

  \foreach \i [count = \ct] in {2,1,3}              %first permutation
    \draw[->,ultra thick,draw=red!70] (\ct 1)--(\i 2);
  \foreach \i [count = \ct] in {1,3,2}              %second permutation
    \draw[->,ultra thick,draw=olive] (\ct 2)--(\i 3);
  \node[draw] at (9,0) {$(1\ \ 2)$};
  \node[draw] at (15,0) {$(2\ \ 3)$};

\end{tikzpicture}

\begin{tikzpicture}[mystyle/.style={circle,draw,fill=gray!40}]
  \foreach \i in {1,...,3}
    \foreach \j in {1,2} 
      {\pgfmathtruncatemacro{\label}{(\i)}
      \node[mystyle]  (\i\j) at (12*\j,-1*\i) {\label};}

  \foreach \i [count = \ct] in {3,1,2}
    \draw[->,ultra thick,draw=blue!70] (\ct 1)--(\i 2);
  \node[draw] at (18,-1) {$(1\ \ 2)(2\ \ 3)$};

\end{tikzpicture}