Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
can be interpreted as the rotation matrix in the complex x 4 x 1 plane (Problem
12.16).
It is straightforward to generalize the above discussion to the general case
where the relative velocity is in an arbitrary direction. The transformation matrix
will be a 4 4 matrix, instead of a 2 2 matrix one. For this general case, we have
to take x 2 - and x 3 -axes into consideration.
Problems
12.1. Show that
(a) the unit element (the identity) in a group is unique, and
(b) the inverse of each group element is unique.
12.2. Show that the set of complex numbers 1; i; 1, and i form a group of
order four under multiplication.
12.3. Show that the set of all rational numbers, the set of all real numbers,
and the set of all complex numbers form in®nite Abelian groups under
addition.
12.4. Show that the four matrices
~A ˆ 1 0
0 1
; ~B ˆ
0
1
1 0
; ~C ˆ 1 0
0 1
form an Abelian group of order four under multiplication.
12.5. Show that the set of all permutations of three objects
‰1 23Š; ‰2 31Š; ‰3 12Š; ‰1 32Š; ‰3 21Š; ‰2 13Š
; ~D ˆ 0 1
1 0
forms a non-Abelian group of order six, with sequential performance as
the law of combination.
12.6. Given two elements A and B subject to the relations A 2 ˆ B 2 ˆ E (the
identity), show that:
(a) AB 6ˆ BA, and
(b) the set of six elements E; A; B; A 2 ; AB; BA form a group.
12.7. Show that the set of elements 1; A; A 2 ; ...; A n1 , A n ˆ 1, where A ˆ e 2i=n
forms a cyclic group of order n under multiplication.
12.8. Consider the rotations of a line about the z-axis through the angles
=2;;3=2; and 2 in the xy plane. This is a ®nite set of four elements,
the four operations of rotating through =2;;3=2, and 2. Show that
this set of elements forms a group of order four under addition.
12.9. Construct the group multiplication table for the group of Problem 12.2.
12.10. Consider the possible rearrangement of two objects. The operation E p
leaves each object in its place, and the operation I p interchanges the two
objects. Show that the two operations form a group that is isomorphic to
G 2 .
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