user26390: points on curve secp256k1 form a group E(Fp) over field Fp.
p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1  is prime.
n is the order of group E, 
n=115792089237316195423570985008687907852837564279074904382605163141518161494337
Is n prime too?
Is E(Fp) a cyclic group?
Theorem. 
Working over a finite field, the group of points E(Fp) is always either a cyclic group or the product of two cyclic groups.

user26915: The comment stating n “has to be prime” is a bit confusing.
The order of base point “has” to be prime in the sense that this is a requirement in the particular documents defining standard curves—for example, in SECG, which includes secp256k1. Bitcoin's base point order r is prime.
In SECG, it is also stated that cofactor of secp256k1 curve is 1, which makes n = r × 1, again prime. A group of prime order is obviously cyclic.

26390: <p>points on curve secp256k1 form a group E(Fp) over field Fp.</p>

<p>p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1  is prime.</p>

<p>n is the order of group E, 
n=115792089237316195423570985008687907852837564279074904382605163141518161494337</p>

<p>Is n prime too?</p>

<p>Is E(Fp) a cyclic group?</p>

<p>Theorem. 
Working over a finite field, the group of points E(Fp) is always either a cyclic group or the product of two cyclic groups.</p>
 26915: <p>The comment stating <code>n</code> “has to be prime” is a bit confusing.</p>

<p>The order of base point “has” to be prime in the sense that this is a requirement in the particular documents defining standard curves—for example, in <a href="http://www.secg.org/sec2-v2.pdf" rel="noreferrer">SECG</a>, which includes secp256k1. Bitcoin's base point order <code>r</code> is prime.</p>

<p>In SECG, it is also stated that cofactor of secp256k1 curve is 1, which makes <code>n = r × 1</code>, again prime. A group of prime order is obviously cyclic.</p>
