Abel's theorem in problems and solutions - School of Mathematics

122 Problems of Chapter 1
contradiction Hence Since is an arbitrary element
different from also From the
equality it follows that and
The table of multiplication is now uniquely defined. Indeed,
Hence in this case we can have only one group. We
can verify that our multiplication table in fact defines a group. This group
is called the group of quaternions. It is better to denote its elements by
the following notations: instead of we have, in
that order, Thus the multiplication by 1 and
–1 and the operations with signs are the same as in ordinary algebra.
Moreover, one has
The multiplication table for the group of quaternions is
shown in Table 11.
93. Consider the vertex named A by the new notation. Its old notation
was thus By the action of the given transformation this
vertex is sent onto a vertex named, in the old notation, and, in
the new notation, Similarly in the new notations the vertex
B is sent onto and the vertex C onto Hence to this
transformation there corresponds in the new notation the permutation
94. if and only if Hence every element has,
under the mapping one and only one image. It follows that
is a bijective mapping of the group into itself. Moreover,
Thus
is an isomorphism.
95. Answer. The reflections with respect to all altitudes.
96. Answer. The rotations by 120° and 240°.
97. Answer. Let us subdivide all the elements of the group of symmetries
of the tetrahedron in the following classes: 1) 2) all rotations
different from about the altitudes; 3) all rotations by 180° about the
axes through the middle points of opposite edges; 4) all reflections with
respect to the planes through any vertex and the middle point of the opposite
edge; 5) all transformations generated by a cyclic permutation of
the vertices (for example, ). Thus two elements can be
transformed one into another by an internal automorphism of the group