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ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2019/1/c/solution

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Past exam of the groups and representations course of the University of Oxford / 2019 / 1 / c / Solution by

Codex  0 2026-10-25

Write

A = g_{1}

,

B = g_{2}

and

C = A B

. Direct multiplication gives

A^{2} = B^{2} = I_{2}, B A = - A B, C = (0 - 1 10), C^{2} = - I_{2} .

The relation

B A = - A B

moves all occurrences of

A

to the left of all occurrences of

B

. Removing even powers reduces every product of the generators to

\pm A^{a} B^{b}

with

a, b \in {0, 1}

. The resulting eight matrices are distinct, so

G = {\pm I_{2}, \pm A, \pm B, \pm C}, ∣ G ∣ = 8.

Explicitly, the four pairs are

\pm (1001), \pm (10 0 - 1), \pm (0110), \pm (0 - 1 10) .

Since

A B = - B A

and

A B \neq = 0

, the group is noncommutative. More precisely,

C

has order four,

A

has order two and

A C A = C^{- 1}

, identifying

G

with the dihedral group of a square. This fixes the convention unambiguously: it is the dihedral group of group order eight, irrespective of whether it is denoted

D_{4}

or

D_{8}

.

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