About the definition of Sylow p-groups

Question

Here is the definition of Sylow p-group (source: wikipedia)

For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e. a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).

Does that mean it is possible that if I have $2$ sylow p-groups $P_1,P_2$ with $P_1 \neq P_2$ then

$|P_1| \neq |P_2|$ (if neither $P_1 \leq P_2,$ nor $P_2 \leq P_1)$ or am I misunderstanding the definition.

Answer 1

The definition would a priori allow to have $|P_1|\ne|P_2|$. However, the Sylow theorems tell us that the Sylow p-subgroups are conjugates of each other and hence of same order (in fact, isomorphic).

Answer 2

Some authors define Sylow $p$-subgroups having order $p^k$ with $k$ maximal, see here:

Definition: If $p^k$ is the highest power of a prime $p$ dividing the order of a finite group $G$, then a subgroup of G of order $p^k$ is called a Sylow $p$-subgroup of $G$.

With this definition, $|P_1|=|P_2|$. There is also another definition as a maximal $p$-subgroup of $G$, and then we would use a Sylow Theorem to show that $P_1=gP_2g^{-1}$ for some $g$, so that $|P_1|=|P_2|$.