Subgroups of Cyclic Groups

Abstract Algebra · Axiom Academy

LESSON Subgroups of Cyclic Groups Every subgroup of a cyclic group is itself cyclic, and the lattice of subgroups corresponds beautifully to the divisors of the group's order. 1. Cyclic Groups and Generation We write and call a generator of the group. Example: Z₆ under addition mod 6 The element 1 generates all of Z₆: Proof Sketch: Let be a subgroup of . If is trivial, it's cyclic. Otherwise, let be the smallest positive integer such that is in . We can show that , making cyclic. For a cyclic group of order , there's a perfect correspondence: If generates and divides , then the unique subgroup of order is: The divisors of 12 are: 1, 2, 3, 4, 6, 12 4. The Subgroup Lattice of Z₃₀ The subgroups of a cyclic group form a lattice under set inclusion. For Z₃₀, since 30 = 2 × 3 × 5, the divisors are: 1, 2, 3, 5, 6, 10, 15, 30. In the lattice above, each node represents a subgroup, labeled by its order. An upward path from subgroup H to subgroup K means H ⊆ K, which happens precisely when |H| divides |K|. ⟨6⟩ = 0, 6, 12, 18, 24 has order 5 ⟨5⟩ = 0, 5, 10, 15, 20, 25 has order 6 Notice: ⟨15⟩ ⊆ ⟨5⟩ because 2 | 6, and ⟨10⟩ ⊆ ⟨5⟩ because 3 | 6.

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