can composite groups have primitive roots (be cyclic)?

Question

Imagine Z/nZ with n not prime (n=pq). Can the multiplicative group be cyclic?

I read the paper over here: http://alpha.math.uga.edu/~pete/4400primitiveroots.pdf

At one point he dismiss the case where the group can be isomorphic to the product of several groups:

Thus it is enough to figure out the group structure when $N=p^a$ is a prime power.

Is it because a product of group can't be cyclic?

Thanks!

score 1 · Answer 1 · answered May 02 '16 at 18:30

1

Yes, this can happen for $p=2$ and $q>2$ another prime. In fact, there is a primitive root modulo $n$ if and only if $n$ is $1$ or $2$ or $4$ or $p^\alpha$ or $2p^\alpha$, where $p$ is prime, $p\ne2$, and $\alpha\ge1$, see Corollary $2$ of P. Clark's text. For $n=pq$ with $p>2,q>2$ both prime there is hence no primitive root, so that $(\mathbb{Z}/pq)^*$ is not cyclic in this case.

answered May 02 '16 at 18:30

Dietrich Burde

130,978

but why can't a product of two prime groups can't be cyclic? – David 天宇 Wong May 02 '16 at 18:42
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Because of an analysis of the order of elements. This is given in all detail in the notes of Pete Clark, i.e., the proof of Theorem $1$. – Dietrich Burde May 02 '16 at 19:03
OK I understand now! What about this "their orders are not relatively prime and the product group cannot be cyclic". Why is that not allowing the product to be cyclic? – David 天宇 Wong May 02 '16 at 19:06
found a question about that here: http://math.stackexchange.com/questions/5969/product-of-two-cyclic-groups-is-cyclic-iff-their-orders-are-co-prime – David 天宇 Wong May 02 '16 at 19:09

score 1 · Answer 2 · edited Apr 13 '17 at 12:19

1

OK ended up finding why:

if $(Z_n)^\ast \sim (Z_p)^\ast \times (Z_q)^\ast$ (the simplest case with $p$ and $q$ primes)

then the order of $(Z_p)^\ast$ is $p-1 = 2 \times \cdots$,

and the order of $(Z_q)^\ast$ is $q-1 = 2 \times \cdots$,

this means the gcd of their order will not be equal to 1, this means the product group won't be cyclic. Explanation about that part here

edited Apr 13 '17 at 12:19

Community

1

answered May 02 '16 at 19:05

David 天宇 Wong

195
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can composite groups have primitive roots (be cyclic)?

2 Answers2