Associative Axiom

Let

(1)
\begin{pmatrix} a & b \\ c & d \end{pmatrix}, \begin{pmatrix}e & f \\ g & h \end{pmatrix}, and \begin{pmatrix}i & j \\ k & l \end{pmatrix}

be elements of GL2(R).

(2)
\begin{align} (\begin{pmatrix}a & b \\ c & d \end{pmatrix} \begin{pmatrix}e & f \\ g & h \end{pmatrix}) \begin{pmatrix}i & j \\ k & l \end{pmatrix}= \begin{pmatrix}ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix} \begin{pmatrix}i & j \\ k & l \end{pmatrix}= \begin{pmatrix}iae+ibg+kaf+kbh & jae+jbg+laf+lbh \\ ice+idg+kcf+kdh & jce+jdg+lcf+ldh \end{pmatrix} \end{align}

And

(3)
\begin{pmatrix} a & b \\ c & d \end{pmatrix} (\begin{pmatrix}e & f \\ g & h \end{pmatrix} \begin{pmatrix}i & j \\ k & l \end{pmatrix})= \begin{pmatrix}a & b \\ c & d \end{pmatrix} \begin{pmatrix}ei+fk & ej+fl \\ gi+hk & gj+hl \end{pmatrix} = \begin{pmatrix}aei+afk+bgi+bhk & aej+afl+bgj+bhl \\ cei+cfk+dgi+dhk & cej+cfl+dgj+dhl \end{pmatrix}

Since R is commutative

(4)
\begin{pmatrix} iae+ibg+kaf+kbh & jae+jbg+laf+lbh \\ ice+idg+kcf+kdh & jce+jdg+lcf+ldh \end{pmatrix}= \begin{pmatrix}aei+afk+bgi+bhk & aej+afl+bgj+bhl \\ cei+cfk+dgi+dhk & cej+cfl+dgj+dhl \end{pmatrix}

By substitution

(5)
\begin{align} (\begin{pmatrix}a & b \\ c & d \end{pmatrix} \begin{pmatrix}e & f \\ g & h \end{pmatrix}) \begin{pmatrix}i & j \\ k & l \end{pmatrix}= \begin{pmatrix}a & b \\ c & d \end{pmatrix} (\begin{pmatrix}e & f \\ g & h \end{pmatrix} \begin{pmatrix}i & j \\ k & l \end{pmatrix}) \end{align}

Therefore, My group is associative

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