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A SU(2) representation descends to the SO(3) group exactly when $-I$ acts trivially. In ${\mathrm{Sym}}^n({\mathbb{C}}^2)$ it acts by $(-1{)}^n$, so descent holds precisely for even $n$, or integer spin $j=n/2$. This is the inflation of a group representation criterion for the kernel $\{\pm I\}$. Half-integer-spin representations instead remain representations of the covering group.

The commutator subgroup, or derived subgroup, isA linear character is afforded by a one-dimensional representation $\rho :G\to {\mathbb{C}}^{\times }$. Since ${\mathbb{C}}^{\times }$ is abelian, every commutator lies in $\ker \rho$, so

Let $\pi :G\to G/G^{\prime }$ be the quotient map. Every irreducible representation of a finite abelian group is one-dimensional, so each irreducible character $\psi$ of the abelianization $G/G^{\prime }$ inflates to the linear character $\psi \circ \pi$ of $G$. Conversely, the containment $G^{\prime }\le \ker \chi$ makes every linear character of $G$ factor uniquely through $G/G^{\prime }$. Thus inflation of a group representation gives a bijection