Exercise 2.3.6 (Lattice of the subgroups of $\mathbb{Z}/48\mathbb{Z}$)

Proof. The divisors of $48=2^4\cdot 3$ are $1,2,4,8,16,3,6,12,24,48$. As in example 1, the subgroups of $\mathbb{Z}∕48\mathbb{Z}$ are

• order 48: $⟨1⟩=⟨5⟩=⟨7⟩=⟨11⟩=⟨13⟩=⟨17⟩=⟨19⟩=⟨23⟩=⟨25⟩=⟨29⟩=⟨31⟩=⟨35⟩=⟨37⟩=⟨41⟩=⟨43⟩=⟨47⟩$
• order 24: $⟨2⟩=⟨10⟩=⟨14⟩=⟨22⟩=⟨26⟩=⟨34⟩=⟨38⟩=⟨46⟩$
• order 16: $⟨3⟩=⟨9⟩=⟨15⟩=⟨21⟩=⟨27⟩=⟨33⟩=⟨39⟩=⟨45⟩$
• order 12: $⟨4⟩=⟨20⟩=⟨28⟩=⟨44⟩$
• order 8: $⟨6⟩=⟨18⟩=⟨30⟩=⟨42⟩$
• order 6: $⟨8⟩=⟨40⟩$
• order 4: $⟨12⟩=⟨36⟩$
• order 3: $⟨16⟩=⟨32⟩$
• order 2: $⟨24⟩$
• order 1: $⟨0⟩$

Note that there are $\phi (d)$ elements of order $d$ for every divisor $d$ of $48$.

The lattice of divisors of $48=2^4$ for divisibility is given by

By Theorem 7, this lattice is isomorphic to the lattice of subgroups of $\mathbb{Z}∕48\mathbb{Z}$ for inclusion (inclusions go from bottom to top):