True or false: "The order of elements in a group with 30 elements can be any number from 1 to 30."

False — by Lagrange's theorem, element orders must divide the group order. In a group of order 30, only divisors of 30 are possible orders.

Divisors of 30: $\{1, 2, 3, 5, 6, 10, 15, 30\}$ — only 8 possible orders, not 30.

Impossible orders: 4, 7, 8, 9, 11, 12, 13, 14, 16-29 — these CANNOT occur as element orders.

Lagrange's theorem: The order of any element (and any subgroup) must divide the group order. This is one of the most fundamental results in group theory.

Important nuance: Not all divisors necessarily occur as element orders — Lagrange only says orders must divide $|G|$, not that every divisor appears. For example:

• The group might have no element of order 30 (then it's not cyclic)
• If it IS cyclic, then for every divisor $d$ of 30, there are exactly $\varphi(d)$ elements of order $d$

Tip: This comes up constantly in crypto parameter selection — when working in $\mathbb{Z}_p^*$ with $|G| = p - 1$, the possible subgroup orders are exactly the divisors of $p - 1$. Choosing $p$ such that $p - 1$ has a large prime factor ensures a large prime-order subgroup for DH.

Go deeper:

• Lagrange's theorem (Wikipedia) — why element orders are restricted to divisors of |G|.