Subgroup Lattice Generator
Subgroup Lattice Generator computes and visualizes the lattice of subgroups for the cyclic group \\(\mathbb{Z}/n\mathbb{Z}\\). Enter a positive integer \\(n\\) to see all subgroups, their orders, generators, and the lattice structure.
Subgroup Lattice Overview
For a cyclic group \\(\mathbb{Z}/n\mathbb{Z}\\) of order \\(n\\), the subgroups correspond to the divisors \\(d\\) of \\(n\\). Each subgroup has order \\(n/d\\) and is generated by \\(a = d\\). The lattice shows the inclusion relations, where a subgroup of order \\(n/d_1\\) is contained in a subgroup of order \\(n/d_2\\) if \\(d_2\\) divides \\(d_1\\).
Definition: A subgroup of \\(\mathbb{Z}/n\mathbb{Z}\\) generated by \\(a = n/d\\) has order \\(d\\), where \\(d\\) divides \\(n\\).
Property: The number of elements in \\(\mathbb{Z}/n\mathbb{Z}\\) of order \\(d\\) is \\(\phi(d)\\), where \\(\phi\\) is the Euler Totient Function.
Lattice: Edges connect subgroups \\(H \subseteq K\\) if \\(K\\) contains \\(H\\), i.e., if the order of \\(H\\) divides the order of \\(K\\).
Example Calculations
Example 1: \\(n = 6\\)
Divisors: \\(1, 2, 3, 6\\).
Subgroups: \\(\langle 6 \rangle = \{0\}, \langle 3 \rangle = \{0, 3\}, \langle 2 \rangle = \{0, 2, 4\}, \langle 1 \rangle = \{0, 1, 2, 3, 4, 5\}\\).
Orders: \\(1, 2, 3, 6\\).
Lattice: \\(\{0\} \subseteq \{0, 3\} \subseteq \{0, 1, 2, 3, 4, 5\}\\), \\(\{0\} \subseteq \{0, 2, 4\} \subseteq \{0, 1, 2, 3, 4, 5\}\\).
Example 2: \\(n = 12\\)
Divisors: \\(1, 2, 3, 4, 6, 12\\).
Subgroups: \\(\langle 12 \rangle = \{0\}, \langle 6 \rangle = \{0, 6\}, \langle 4 \rangle = \{0, 4, 8\}, \langle 3 \rangle = \{0, 3, 6, 9\}, \langle 2 \rangle = \{0, 2, 4, 6, 8, 10\}, \langle 1 \rangle = \{0, 1, …, 11\}\\).
Orders: \\(1, 2, 3, 4, 6, 12\\).
\\(\phi(12) = 4\\) (elements of order 12: \\(1, 5, 7, 11\\)).