Let 
be the group of rigid motions (i.e., orientation preserving isometries) of a tetrahedron in 
. Show that 
.
For reference, consider the following diagram of a tetrahedron.
Let
be an orientation-preserving isometry of the tetrahedron; that is, if the vertices of a face, read clockwise from outside the figure, are 
, then 
are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 4 possibilities for
. Once is chosen, there are 3 possibilities for 
. Now once and are chosen, 
is determined by orientation, and there is only one possibility remaining for 
. Thus there are 
total possibilities for , all distinct. Hence .
be the group of rigid motions (i.e., orientation preserving isometries) of a tetrahedron in . Show that .For reference, consider the following diagram of a tetrahedron.
Let
be an orientation-preserving isometry of the tetrahedron; that is, if the vertices of a face, read clockwise from outside the figure, are , then are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.There are 4 possibilities for
. Once is chosen, there are 3 possibilities for . Now once and are chosen, is determined by orientation, and there is only one possibility remaining for . Thus there are total possibilities for , all distinct. Hence .
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