Physics 227 Lecture 10 Appendix B 1 Autumn 2007
Lecture 10 – Appendix B: Some sample problems from Boas
Here are some solutions to the sample problems assigned for Chapter 3.13.
§3.13: 6 We consider the group formed by the integers 0,1,2,3 with multiplication
defined by addition mod 4, e.g.,
1 2 1 2 3,3 2 5 mod 4 1
.
Solution: Using this rule we obtain the following multiplication table,
0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2
We recognize this table as being of order 4 (only 4 elements) and essentially the same
as (isomorphic to) the tables in Eqs. 13.1 and 13.2 in Boas. Thus this group is
isomorphic to the cyclic group of order 4.
§3.13: 11 Consider the symmetry group of the rectangle, like the square but with
unequal sides.
Solution: Unlike the case of a square the rotations through 90 degrees (270 degrees)
and the reflections through the diagonals are no longer symmetries. We are left with
just the following transformations, the identity, the rotation through 180 degrees, the
reflection through the x-axis (change the sign of y), and the reflection through the y-
axis (change the sign of x). These can be represented by the following 2x2 matrices
1 0 1 0 1 0 1 0
1 , 1, , ,
0 1 0 1 0 1 0 1
x y
R P P
with multiplication table
