Course 111: Algebra, 3rd November 2006
1. Consider x1 , x2 , . . . , xn elements of a group G. Then products of the
group elements can be written as
x1 x2 x3
x1 x2 x3 x4
x1 x2 x3 . . . x m
= (x1 x2 )x3
= (x1 x2 x3 )x4 = ((x1 x2 )x3 )x4
···
= (x1 x2 . . . xn−1 )xn = (. . . ((x1 x2 )x3 ) . . . xn−1 )xn
The general associative law determines that the value of a product of
n group elements depends only on the order in which those elements
occur within the product.
Eg. Consider 4 elements x1 , x2 , x3 and x4 . The expressions (x1 x2 )(x3 x4 ),
(x1 (x2 x3 ))x4 , x1 ((x2 x3 )x4 ) and x1 (x2 (x3 x4 )) all result in the same value.
Show how the general associative law can be verified by induction on
the number of elements involved.
Note: the only group axiom you should need is the associative law for
products of three elements.
Proof:
See David Wilkins on-line Group theory notes, page 6 at
http://www.maths.tcd.ie/ dwilkins/Courses/111/
for a nice presentation of this proof.
2. Enumerate the rotations and reflections of a square which leave it invariant and determine the Cayley table for this symmetry group.
Note: the symmetry group, S4 which is the permutation group of four
objects - the four vertices in this case, is not the same as the symmetry
group for the square, D8 (or sometimes written D4 ).
Consider the rotations of a cube. Show, by listing them, that there are
24 such symmetries.
Symmetries of the square
There is a very nice web page that discusses the symmetries of the
square and the cube at
http://www.maths.uwa.edu.au/ schultz/3P5.2000/3P5.2,3SquareCube.html
There you will see the 8 symmetries of the square given by
and enumerated as:
• E .. This is the ’stay-put’ transformation. E is called the identity.
• U .. This rotates the square through a quarter turn.
• V .. This rotates the square through a half turn.
• W .. This rotates the square through a three-quarters turn.
• P .. This reflects about a vertical mirror line through the centre.
• Q .. This reflects about a horizontal mirror line through the centre.
• R .. This reflects about a diagonal mirror line through AC.
• S .. This reflects about a diagonal mirror line through BD.
with a Cayley table given by
E
U
V
W
P
Q
R
S
E
E
U
V
W
P
Q
R
S
U
U
V
W
E
R
S
Q
P
V
V
W
E
U
Q
P
S
R
W P
W P
E S
U Q
V R
S E
R V
P W
Q U
Q
Q
R
P
S
V
E
U
W
R
R
P
S
Q
U
W
E
V
S
S
Q
R
P
W
U
V
E