Question

Show: all SU(2) matrices satisfy the 4 defining properties of a group.

Answer 1

SU(2) is special unitary group of degree 2. The group operation is matrix multiplication. Every matrix must have determinant 1. Entries can be complex.

Properties of group are:

1) Closure: If a and b belong to group than a.b belong to the group.

2) associativity: if a,b,b belong to the group than (a.b).c=a.(b.c)

3) identity: there exist an element e such that a.e=e.a=a holds true for every a in the group.

4) inverse: for every a there exist an a^-1 such that a.a^-1 =a^-1 .a=e

For SU(2)

i) Closure: If A and B belong to SU(2) than det(A)=det(B)=1. if A.B=C than det(C)=det(A.B)=det(A)det(B)=1

therefore A.B belongs to the group.

ii) Associativity: Matrix multiplication is associative..

iii)Identity: Identity matrix is the identity element. Let's call it E.

iv) Inverse: As the matrix is not singular there will be a inverse for every matrix. What we need is that the inverse must belong to the group. If matrix A has inverse A^-1 than A.A^-1 =E.

det(E)=1=det(A.A^-1)=det(A)det(A^-1)

which implies that 1=1*det(A^-1). Therefore A^-1 belongs to the group SU(2).