The Pauli matrices satisfy for and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

The Pauli matrices satisfy

\sigma_a^2=\mathsf{1}

for

a=x,y,z

and

\sigma_x\sigma_y=i\sigma_z=-\sigma_y\sigma_x

, with the same relations for cyclic permutations of

x

y

and

z

. Let

Q=\{\pm\mathsf{1},\ \pm i\sigma_x,\ \pm i\sigma_y,\ \pm i\sigma_z\}

, with matrix multiplication as the product. ((a)) Show that

Q

is a group of order 8 and that it is not abelian. ((b)) Find the order of each of the eight elements of

Q

. ((c)) The group

C_{4v}

of a square-based pyramid (the rotations

E

C_4

C_2=C_4^2

C_4^3

about the axis and four reflections in vertical planes) also has order 8 and is not abelian. Decide whether

Q\cong C_{4v}

, and give the reason.