Moderate# Quotient of a Group by its Derived Subgroup: Basic Properties

ABSALG-LVKIXI

Let $G$ be a group and $G':=G^{(1)}:=[G,G]$ the commutator or derived subgroup of $G$.

Recall that $[G,G]$ is the subgroup of $G$ generated by the commutators $[g,h]=ghg^{-1}h^{-1}$, where $g,h\in G$.

We have $[G,G]\trianglelefteq G$ (that is, $[G,G]$ is a normal subgroup of $G$), and we define the ** abelianization** $G^{{\rm ab}}$ of $G$ to be the quotient group $G^{{\rm ab}}=G/[G,G]$.

Recall that a group $H$ is abelian if and only if the commutator of any two elements equals the identity $e$ of $H$.

For the statement in each part of this question, answer **True** if it is always true, or **False** if it is sometimes false.