Groups — Definition, Examples, Subgroups

GRE Math Subject · Axiom Academy

LESSON Groups — Definition, Examples, Subgroups The foundational algebraic structure: sets with a single operation A group is a set with a binary operation satisfying: If for all , the group is abelian (or commutative). — integers under addition (identity: 0, inverse of is ) — integers modulo (residue classes) — units modulo (invertible elements) — all bijections from a set to itself, composition — even permutations (alternating group) (trivial subgroup) and itself (improper subgroups) The order is the number of elements in . The order of an element is the smallest positive integer such that . An element is a generator if (the subgroup generated by is all of ). GRE Tip: Identify generators and subgroups quickly. Finite abelian groups are the most common test structures.

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