\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

Groups. Group datum.

The datum of a group can be represented completely by a table that meets certain criteria. For a group \( G \) with set \( \{e_G, a, b, c, d, x, y, z \} \) and operation \( \cdot_G \), the table below is a scaffold for the multiplication table for \( \cdot_G \). The row and column corresponding to the identity is filled in, as the values are know from the definition of a group.

Table for \( \cdot_G \)
	\( e_G \)	a	b	c	d	x	y	z
\( e_G \)	\( e_G \)	a	b	c	d	x	y	z
a	a
b	b
c	c
d	d
x	x
y	y
z	z

Sufficiency

The table has enough information to record all aspects of group \( G \), and in particular, it records sufficient information to confirm that the group meets conditions such has having an identify and the presence of element inverses.

The reverse side lists an interesting property of the rows and columns of the table. Can you remember it? [Dummy cloze]