So let's let
<ul>
<li>
<b>e</B> stand for the <b>Do nothing</B> movement.</li>

<li>
<b>a</B> stand for the <b>Rotate 120 degrees counterclockwise</B> 
movement.</li>

<li>
<b>b</B> stand for the <b>Rotate 240 degrees counterclockwise</B>
movement.</li>

<li>
<b>X</B> stand for the <b>Flip about axis through the top vertex</B> 
movement.</li>

<li>
<b>Y</B> stand for the <b>Flip about the axis through the lower-left 
vertex</B> movement.</li>

<li>
<b>Z</B> stand for the <b>Flip about the axis through the lower-right 
vertex</B> movement.</li>
</ul>
<p>

With these definitions for <b>e, a, b, X, Y,</B> and <b>Z</B> we can
form the Cayley table for the symmetry group of the equilateral triangle:
(In this table the first motion performed is in the left-hand column 
and the second motion is in the top row.)
<p>

<TABLE BORDER>
<TR>
<TD> &#149 </TD><TD> e </TD><TD> a </TD><TD> b </TD>
            <TD> X </TD><TD> Y </TD><TD> Z </TD>
</TR>
<TR>
<TD> e </TD><TD> e </TD><TD> a </TD><TD> b </TD>
            <TD> X </TD><TD> Y </TD><TD> Z </TD>
</TR>
<TR>
<TD> a </TD><TD> a </TD><TD> b </TD><TD> e </TD>
            <TD> Y </TD><TD> Z </TD><TD> X </TD>
</TR>
<TR>
<TD> b </TD><TD> b </TD><TD> e </TD><TD> a </TD>
            <TD> Z </TD><TD> X </TD><TD> Y </TD>
</TR>
<TR>
<TD> X </TD><TD> X </TD><TD> Z </TD><TD> Y </TD>
            <TD> e </TD><TD> b </TD><TD> a </TD>
</TR>
<TR>
<TD> Y </TD><TD> Y </TD><TD> X </TD><TD> Z </TD>
            <TD> a </TD><TD> e </TD><TD> b </TD>
</TR>
<TR>
<TD> Z </TD><TD> Z </TD><TD> Y </TD><TD> X </TD>
            <TD> b </TD><TD> a </TD><TD> e </TD>
</TR>
</TABLE>