What is a subgroup, what is the order of an element, and how do they relate to the group order?

A subgroup is a subset that is itself a group under the same operation. The order of an element $a$ is the smallest $m$ where $a^m = e$. Both must divide the group order (Lagrange's theorem).

Subgroup: A subset $U \subseteq G$ that is closed under the group operation (all other group properties follow automatically).

Order of an element: $\text{ord}(a) = m$ where $m$ is the smallest positive integer with $a^m = e$ (neutral element).

• For additive groups: $m \cdot a \equiv 0 \mod n$
• For multiplicative groups: $a^m \equiv 1 \mod n$

Lagrange's theorem: The order of any subgroup (and any element) must divide the group order.

Example: $\mathbb{Z}_7^* = \{1, 2, 3, 4, 5, 6\}$, $\cdot \mod 7$, $|G| = 6$

Possible element orders must divide 6: $\{1, 2, 3, 6\}$

Key fact: $a^{\text{ord}(a)-1} \equiv a^{-1} \mod n$ — the inverse can be computed from the element's order.

Go deeper:

• Lagrange's theorem (Wikipedia) — why subgroup and element orders must divide |G|.