 Solution

The four displayed matrices fail closure: for example ${\sigma }_x{\sigma }_y=i{\sigma }_z$ is not one of them. Thus they do not form a group by themselves.

Let ${\sigma }_0=I$ and ${\mu }_4=\{1,-1,i,-i\}$. The enlarged set is the single-qubit Pauli groupThere are 16 distinct matrices: $I,{\sigma }_x,{\sigma }_y,{\sigma }_z$ are linearly independent, so two different ones cannot be scalar multiples, and the four phases are distinct for each nonzero matrix.

The Pauli matrices satisfyHere $\delta$ is the Kronecker delta and $ϵ$ the Levi-Civita symbol. Thus any product of two Pauli matrices is another Pauli matrix or the identity times a phase in ${\mu }_4$. Multiplying by the two original phases preserves membership in ${\mu }_4$, proving closure of all 16 matrices.

Associativity is inherited from matrix multiplication, the identity is $I$, andThis proves every group axiom. The resulting group is non-Abelian, since reversing ${\sigma }_x{\sigma }_y$ changes its sign.