 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/10/b/solution

The products in (a), together with

(- A) B = A (- B) = - (A B)

and

(- A) (- B) = A B

, give closure of the eight-element set. Associativity is inherited from matrix multiplication,

W

is the identity element, and the inverses from (a), with

(- A)^{- 1} = - A^{- 1}

, remain in the set. Thus the group axioms hold. This is the quaternion group

Q_{8}

In the element order specified in the question, the multiplication table is

W X Y Z W X Y Z W W X Y Z W X Y Z X X W Z Y X W Z Y Y Y Z W X Y Z W X Z Z Y X W Z Y X W W W X Y Z W X Y Z X X W Z Y X W Z Y Y Y Z W X Y Z W X Z Z Y X W Z Y X W

Interpreting “character of the group” as the character of a representation specified by these matrices, take the trace of each matrix:

χ (W) = 2, χ (W) = - 2, χ (X) = χ (Y) = χ (Z) = χ (X) = χ (Y) = χ (Z) = 0.

A character belongs to a representation; the underlying abstract group can have other characters.

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