Sylow p- subgroups

Question

I am new to the theory of "Sylow p-subgroups", and I am stuck at one point.

My book gives the following definition of a Sylow p-subgroup:

Let $G$ be a group of order $p^km$ where $k>=1$ and p is some prime number such that p does not divide m.Then a subgroup of $G$ of order $p^k$ is called a Sylow p-subgroup.

What i wanted to ask is, that, is it necessary that in a Sylow p-subgroup,which has an order of $p^k$ every element also has its order as some power of $p$.

For example, in a group $G$ of order $56$ a sylow-7 subgroup means that it is a subgroup of order $7$ but at the same time does it also mean that every element of that group also has an order of $7$?

I am confused.

Answer 1

In the Sylow $p$-subgroup every element will have order a power of $p$ by Lagrange's theorem. In the whole group $G$ we can still have elements of other orders.

Answer 2

No: The product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_3$ has a 2-subgroup containing an element of order 4.