Subgroups and cyclic groups
Objective. Apply subgroup tests and classify cyclic groups.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Subgroups and cyclic groups describe substructures generated within a group. A subset H ⊆ G is a subgroup if it is non-empty, closed under the group operation, and closed under inverses; the one-step subgroup test checks ab⁻¹ ∈ H for all a, b ∈ H. A cyclic group ⟨g⟩ consists of all powers gⁿ of a single generator g, and every subgroup of a cyclic group is itself cyclic. In (ℤ, +), the subgroup generated by 3 is 3ℤ = {…,−6,−3,0,3,6,…}, and every subgroup of ℤ has this form nℤ for some n ≥ 0.
Worked example
Determine the subgroup of (ℤ, +) generated by the element 4, listing its first few positive elements.
- 1The cyclic subgroup generated by 4 consists of all integer multiples of 4.
- 2Closure holds since the sum of multiples of 4 is a multiple of 4.
- 3Inverses hold since −4k is also a multiple of 4 for any integer k.
- 4So the subgroup is 4ℤ = {…, −8, −4, 0, 4, 8, …}.
Answer: 4ℤ = {…, −8, −4, 0, 4, 8, …}
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Start practisingCommon mistake. Students check closure under the operation but forget to verify inverses lie in the subset; {0, 1, 2} under addition mod 5 is closed under addition in some cases but 1+2=3 is not in the set, so it fails closure too, showing both checks matter.
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