Solution

The relations in (a) reduce every word in the generators to

U^{a} V^{b}

, with

0 \leq a < 4

and

b \in {0, 1}

. These eight matrices are distinct: the

b = 0

elements have third diagonal entry

1

and the

b = 1

elements have entry

- 1

, while the four planar blocks are distinct in each family. Their multiplication is

(U^{a} V^{b}) (U^{c} V^{d}) = U^{a + (- 1)^{b} c} V^{b + d},

with exponents reduced modulo four and two. This proves closure. The identity element is

U^{0} V^{0}

, the inverse of

U^{a}

U^{- a}

, and each

U^{a} V

is its own inverse. Associativity is inherited from matrix multiplication, proving the group axioms. This is the eight-element dihedral group

D_{8}

g_{j} = U^{j}

and

g_{j + 4} = U^{j} V

for

j = 0, 1, 2, 3

, consistently with the two labels specified in the question. The row-times-column multiplication table is

g_{0} g_{1} g_{2} g_{3} g_{4} g_{5} g_{6} g_{7} g_{0} g_{0} g_{1} g_{2} g_{3} g_{4} g_{5} g_{6} g_{7} g_{1} g_{1} g_{2} g_{3} g_{0} g_{7} g_{4} g_{5} g_{6} g_{2} g_{2} g_{3} g_{0} g_{1} g_{6} g_{7} g_{4} g_{5} g_{3} g_{3} g_{0} g_{1} g_{2} g_{5} g_{6} g_{7} g_{4} g_{4} g_{4} g_{5} g_{6} g_{7} g_{0} g_{1} g_{2} g_{3} g_{5} g_{5} g_{6} g_{7} g_{4} g_{3} g_{0} g_{1} g_{2} g_{6} g_{6} g_{7} g_{4} g_{5} g_{2} g_{3} g_{0} g_{1} g_{7} g_{7} g_{4} g_{5} g_{6} g_{1} g_{2} g_{3} g_{0}

The rotation powers commute with one another, and

U^{a}

commutes with

U^{c} V

exactly when

a

is even. Two elements

U^{a} V, U^{c} V

commute exactly when

a - c

is even. These statements also check the symmetric portions of the table.

 Ancestors (13)

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