MATH 01340 · Groups

Subgroups, cyclic groups

Definition

H≤G if H nonempty and closed under the operation and inverses (or use one-step subgroup tests). Cyclic group: generated by one element, G=⟨g⟩={gᵏ}.

⟨g⟩ is always abelian. Order of g is smallest positive k with gᵏ=e (or infinite).

Subgroup criterion (finite): nonempty closed subset of a finite group is a subgroup.

Examples: nℤ ≤ ℤ; rotations of a regular n-gon.

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