True or false: "A group with 29 elements is always cyclic."

True — 29 is prime, and every group of prime order is cyclic. Moreover, every non-identity element is a generator.

Why prime order guarantees cyclicity:

• By Lagrange's theorem, the order of every element must divide the group order
• If $|G| = 29$ (prime), the only divisors are 1 and 29
• Every element either has order 1 (the identity) or order 29
• An element of order 29 generates the entire group → cyclic

Properties of this group:

• Number of generators: $\varphi(29) = 28$ — that's every non-identity element
• Only subgroups: $\{e\}$ (order 1) and $G$ itself (order 29) — no proper subgroups exist
• The group is necessarily Abelian (commutative)

Contrast with non-prime order: A group of order 30 could have elements of various orders (divisors of 30: 1, 2, 3, 5, 6, 10, 15, 30). It might be cyclic (if an element of order 30 exists) or non-cyclic.

Cryptographic relevance: This is why elliptic curve groups with prime order are preferred — every non-identity point is a generator, simplifying parameter selection and preventing small-subgroup attacks.

Go deeper:

• Cyclic group (Wikipedia) — prime order always forces a cyclic group.