# Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract Algebra by Dummit and Foote and the following definitions are made:

$$1$$. A group of order $$p^{\alpha}$$ for some $$\alpha\geq1$$ is called a $$p$$-group. Subgroups of $$G$$ which are $$p$$-groups are called $$p$$-subgroups.

$$2$$. If $$G$$ is a group of order $$p^{\alpha}m$$, where $$p$$ does not divide $$m$$, then a subgroup of order $$p^{\alpha}$$ is called a Sylow $$p$$-subgroup of G.

As I see it the difference seems to be that if $$H \leq G$$ with $$|H|=p^{\beta}$$ for $$\beta\geq1$$ it is called a $$p$$-subgroup if $$|G|=p^{\alpha}$$ and a sylow subgroup of $$G$$ if its order is not a power of $$p$$.

But in the wording of the Sylow's theorem in the book it reads that

Let G be a group of order $$p^{\alpha}m$$, where $$p$$ is a prime not dividing $$m$$ then...If $$P$$ is a Sylow $$p$$-subgroup of $$G$$ and $$Q$$ is any $$p$$-subgroup of G...

I don't understand the difference between $$P$$ and $$Q$$..both are subgroups of $$G$$ with order of a power of $$p$$ and I thought that $$Q$$ can't be called a $$p$$ subgroup because the order of $$G$$ is not a power of $$p$$.

So it seems that I didn't understand the difference in definitions $$(1)$$ and $$(2)$$ after all.

• The key point is "Sylow" means maximal prime power factor. If $G$ is a group of order $p^am$ where $m$ is not divisible by $p$, the $p$-subgroups are those with order any power of $p$ (always at most $p^a$) while the Sylow $p$-subgroups are those whose order is $p^a$. – KCd Apr 24 '15 at 21:11

A $$p$$-group is a group where the order of every element is a power of $$p$$. A direct consequence of this is that if the order of every element is a power of $$p$$, then the order of the group must be a power of $$p$$. This follows from Lagrange's theorem: the order of a subgroup divides the order of the group. So, one can say that a $$p$$-group is a group with order a power of $$p$$.
A Sylow $$p$$-subgroup, also called a $$p$$-Sylow subgroup, is a maximal $$p$$-subgroup of a group. This means that it is not a proper subgroup of any other $$p$$-subgroup of the main group. In other words, if $$G$$ is a group with order $$q^{\alpha}m$$, where $$q \nmid m$$, then the first Sylow theorem guarantees us a subgroup with order $$q^{\alpha}$$. This subgroup is called a Sylow $$q$$-subgroup of $$G$$.
One can say that every Sylow $$p$$-subgroup is a $$p$$-group, but not every $$p$$-group is a Sylow $$p$$-subgroup.
Let's take a group $$G$$ with order $$45$$. Then $$\left|G\right|=45=3^2 \cdot 5$$.
A subgroup of $$G$$ with order $$3^2$$ is a Sylow $$3$$-subgroup. A subgroup of $$G$$ with order $$5$$ is a Sylow $$5$$-subgroup of $$G$$. However, a subgroup with order $$3$$ is a $$3$$-subgroup, but it is NOT a Sylow $$3$$-subgroup.