1.1.3 General Linear Group

The gen­er­al linear group con­sists of in­vert­ible linear $n \times n$ ma­tri­ces over the field F with degree $n$, rep­re­sent­ing $GL(n,F)$, for $n\in \mathbb{N}$. If we chose F to be over $\mathbb{R}\ \text{or}\ \mathbb{C}$, there are an in­fi­nite number of pos­si­bil­i­ties, im­ply­ing the group's order will also be in­fi­nite. We find a finite-el­e­ment group when choos­ing F to be finite. Start­ing with the ex­am­ple $GL(2,\mathbb{Z}_{2})$, this rep­re­sents the gen­er­al linear group of order of two, having either 1 or 0 for its matrix en­tries. There are two op­tions $p$, for each of the four en­tries $n$, in each of the group's $2 \times 2$ per­mu­ta­tion ma­tri­ces, to­tal­ing six­teen pos­si­ble per­mu­ta­tions

Its ma­tri­ces must be in­vert­ible from the stand­point of matrix group theory, which re­quires non-zero de­ter­mi­nants. From the six­teen pos­si­ble per­mu­ta­tions, we omit the ten fol­low­ing non-in­vert­ible ma­tri­ces

leav­ing us with the six ma­tri­ces that form the $GL(2,\ \mathbb{Z}_{2})$ group

These six ma­tri­ces $\in GL(2,\mathbb{Z}_{2})$ can be seen as sym­met­ric per­mu­ta­tions of three ob­jects de­not­ed by $S_{3}$. The per­mu­ta­tions of $S_{3}$ are usu­al­ly rep­re­sent­ed as all the pos­si­ble com­bi­na­tions to arrange the let­ters ABC

We will typ­i­cal­ly con­sid­er ma­tri­ces in the book with only be one entry $\in F,$for each row/column pair­ing. The $S_{3}$ sym­me­try group can then be rep­re­sent­ed in a finite group form using the six $3 \times 3$ per­mu­ta­tion ma­tri­ces,

The geo­met­ric in­ter­pre­ta­tion of these ma­tri­ces $\in$ $S_{3}$ can be seen as the number of ways we can rotate the three ver­tices of an equi­lat­er­al tri­an­gle, as seen in the fol­low­ing figure.

A gen­er­al­iza­tion for the number of per­mu­ta­tions ex­pect­ed for the number of per­mu­tat­ing ob­jects $n$, would be equal to $n(n - 1)$. This number co­in­cides with the tri­an­gle's three ver­tices in terms of the $S_{3}$ group, for $n = 3$, and gives the total of $3(3 - 1) = 6$ in­vert­ible per­mu­ta­tions. Taking the set of ma­tri­ces from Eq. 1.1.9, their matrix-mul­ti­pli­ca­tion prod­ucts will then form the finite matrix group rep­re­sen­ta­tion $S_{3}$. An ex­am­ple of the matrix-mul­ti­pli­ca­tion group op­er­a­tion comes by taking the prod­uct $BAC \times BCA$, to find the result $ACB$

We are mind­ful that our $S_{3}$ group sat­is­fies all the group prop­er­ties, having an iden­ti­ty matrix, mul­ti­plica­tive clo­sure, mul­ti­plica­tive as­so­cia­tiv­i­ty, and in­vers­es exists for all the el­e­ments.