# Course 111: Algebra, 3rd November 2006 1. Consider x , x

Course 111: Algebra, 3rd November 2006

1. Consider x

1

, x

2

, . . . , x n elements of a group G . Then products of the group elements can be written as x

1 x

2 x

3

= ( x

1 x

2

) x

3 x

1 x

2 x

3 x

4

= ( x

1 x

2 x

3

) x

4

= (( x

1 x

2

) x

3

) x

4 x

1 x

2 x

3

. . . x m

· · ·

= ( x

1 x

2

. . . x n

1

) x n

= ( . . .

(( x

1 x

2

) x

3

) . . . x n

1

) x n

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 x

1

, x

2

, x

3 and x

4

. The expressions ( x

1 x

2

)( x

3 x

4

),

( x

1

( x

2 x

3

)) x

4

, x

1

(( x

2 x

3

) x

4

) and x

1

( x

2

( x

3 x

4

)) 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.

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, S

4 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, D

8

(or sometimes written D

4

).

Consider the rotations of a cube. Show, by listing them, that there are

24 such symmetries.