Exercise 4.2.3 Edges e, e' of a tetrahedron T are said to be opposite if they are disjoint (that is, they do not share

a vertex). The 6 edges can be partitioned into a set X of three pairs of opposite edges. Prove that G3, the group

of symmetries of T, acts on X and the kernel K< G3 is a normal subgroup of order 4.

The previous exercise is interesting in that it provides a geometric description of a normal subgroup of G3 S₁

different than A₁. It turns out that for any n 4, the only normal subgroup of S₁ is An (and {(1)} and Sn, of

course).