A derived subgroup, also known as the commutator subgroup, is the subgroup generated by all commutators of a group. It serves as a measure of how non-abelian a group is, since the derived subgroup captures the 'failure' of the group to be abelian. This concept is crucial for understanding the structure of groups and analyzing their properties, especially in relation to the derived series, which examines a sequence of derived subgroups to study a group's solvability.
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