Sylow theorems in (Z/pZ)* This might sound as a silly question but does the Sylow theorem III apply to (Z/pZ)*?

For Example for (Z/11Z)* we have the order of the group being 10 = 2 * 5.

I would expect to have only 1 Sylow 5-subgroup. Or am I wrong?
Although not really a question regarding cryptography, Sylow's theorems are quite boring for cyclic groups (or any commutative group):


$(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic and has order $p-1$
Theorem 1 gives you the existance of subgroups with order of every prime power divisor of $p-1$, where every element has a power of that prime as its order.
Theorem 2 on its own isn't all that useful, because conjugation of subgroups doesn't give you a different group in commutative groups: $g^{-1}Hg=g^{-1}gH=H$


Now the part you are asking about: The number of Sylow p-subgroups is fixed - it is $1$. Let's say we want a Sylow p-subgroup for prime $q$ (not to cause confusion with the group order $p-1$, and $q$ divides $p-1$). Theorem 2 states that Sylow groups are conjugate to each other, however conjugation always gives you the same subgroup. Assuming there are two Sylow p-subgroups, then theorem 2 would require them to be conjugate. However, that means those two subgroups are actually equivalent.