% entered by J. Rainbolt and R. Blyth (July 2003)
% edited July 2006
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{{\large \bf Incorporating the Software GAP into Teaching\\ Abstract Algebra
\\PREP Workshop 2006
\\ Section 2: Three Representations of the Group of Rotations of a
Cube
}}
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\noindent
In this series of exercises we consider the group of rotations of the
cube. We start by labeling the vertices of the cube and recording their
movement under various rotations. For example, a rotation of
$90^{\circ}$ about the center of the front face corresponds to the
permutation $(1,2,3,4)(5,6,7,8)$ of the vertices. Likewise, a rotation of
$90^{\circ}$ about the center of the left face gives rise to the
permutation $(1,5,6,2)(3,4,8,7)$.\\
{\tt gap> G8:= SymmetricGroup(8);
Sym( [1 .. 8] )
gap> f8:=(1,2,3,4)(5,6,7,8);
(1,2,3,4)(5,6,7,8)
gap> l8:=(1,5,6,2)(3,4,8,7);
(1,5,6,2)(3,4,8,7)
gap> K8:= Subgroup(G8, [f8,l8]);
Group([(1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7)]);
gap> Order(K8);}\\
\\
Find the effect of following the rotation $f$ by the rotation $l$.\\
{\tt gap> f8*l8;
(2,4,5)(3,8,6)}\\
\\
Note that vertices 1 and 7 are fixed by this rotation of order 3.
Hence the rotation has axis the long diagonal of the cube that runs
through vertices 1 and 7, and is a rotation of $120^{\circ}$\\
\\
{\bf Section 2, Project 1:}\\
\\
2.1 Classify the elements of $K8$ by their types of axes of rotation.\\
\\
2.2 Compute the following products in {\sf GAP}, and give a geometric
description of the rotation (i.e., describe the axis of rotation and
the magnitude of the rotation).\\
a) $f8^{2}$\\
b) $f8^{2}l8^{2}$\\
c) $f8l8f8l8$\\
d) $f8^{2}l8$\\
e) $f8l8f8$\\
\\
Another way to represent the rotations of the cube is to consider the
effect of each rotation on the faces. The rotation $f$ of $90^{\circ}$
about the center of the front face corresponds to the permutation
$(1,5,6,3)$, while the rotation $l$ of $90^{\circ}$
about the center of the left face corresponds to the permutation
$(1,4,6,2)$.\\
{\tt gap> G6:= SymmetricGroup(6);
Sym( [1 .. 6] )
gap> f6:=(1,5,6,3);
(1,5,6,3)
gap> l6:=(1,4,6,2);
(1,4,6,2)
gap> K6:= Subgroup(G6, [f6,l6]);
Group([(1,5,6,3), (1,4,6,2)]);
gap> Order(K6);}\\
\\
{\bf Section 2, Project 2:}\\
\\
2.3 Use this representation to compute the following products in {\sf GAP},
and give a geometric
description of the rotation (i.e., describe the axis of rotation and
the magnitude of the rotation).\\
a) $f6^{3}$\\
b) $f6^{3}l6^{3}$\\
c) $f6l6f6l6f6$\\
d) $f6^{3}l$\\
e) $l6f6l6f6$\\
\\
2.4 Express the rotation $(1,3,4)(2,6,5)$ as a product of $f6$'s and
$l6$'s.\\
\\
2.5 Express the rotation $(1,6)(2,5)(3,4)$ as a product of $f6$'s and
$l6$'s.\\
\\
A third way to represent the rotations of the cube is to consider the
effect of each rotation on the four (long) diagonals.
\\
\\
{\bf Section 2, Project 3:}\\
\\
2.6 Set up the group of rotations of the cube as a subgroup of
$S_{4}$. Again classify the elements of this subgroup by their types of
axes of rotation.
\\
2.7 Use this representation to compute the following products in {\sf GAP},
and give a geometric
description of the rotation (i.e., describe the axis of rotation and
the magnitude of the rotation).\\
a) $l^{2}$\\
b) $l^{2}f^{2}$\\
c) $lflfl$\\
d) $f^{3}l^{2}$\\
e) $flfl^{2}$\\
\\
Note that {\sf GAP} has a command that computes an isomorphism
between two groups if they are isomorphic (and returns ``false'' if
they are not). This works well for small groups.
{\tt gap> IsomorphismGroups(K8,K6);
[ (1,2,3,4)(5,6,7,8), (1,5,6,2)(3,4,8,7) ] -> [ (1,5,6,3), (2,5,4,3) ]}
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