# What is the difference between a $p$-group and a Sylow group?

We have just started a course about Group theory. I am confused about the difference between a $p$-group and a Sylow group.

As I understand it, a group $G$ is called $p$-group if $|G|=p^{m}$, where $p$ is a prime and $0\leq m$.

However, I am not sure what a Sylow group is. I know that there are theorems called sylow theorems that determine the properties of sylow groups, but is there a clear definition of a sylow group and is there a clear difference between sylow group and $p$-group.

• There is no such thing as a Sylow group. They are called Sylow subgroups, and surely your book will have a definition of the term. – Tobias Kildetoft Apr 13 '16 at 12:22
• The word "sub". Sylow subgroup. – user1729 Apr 13 '16 at 12:23
• The Sylow ones have to have the maximum value of $m$ (possible for that $p$). So a group order 72 has Sylow subgroups order 8 and 9. – almagest Apr 13 '16 at 12:25
• @almagest, So every Sylow subgroup is a $p$-group, is that right? – MrDi Apr 13 '16 at 12:30
• Yes, but not every $p$-subgroup is a Sylow subgroup. – almagest Apr 13 '16 at 12:30

A finite $p$-group is a group whose order is a power of $p$.
(More generally, a $p$-group is a group in which each element has order a power of $p$.)
A Sylow subgroup of a group $G$ is a maximal $p$-subgroup of $G$.
Every Sylow subgroup is a $p$-group for some $p$.
Not every $p$-subgroup of a group is a Sylow subgroup. For instance, $C_2 \times 1$ is not a Sylow subgroup of $C_2 \times C_4$.
• Though any $p$-group is a Sylow subgroup of some group (e.g. itself). – Tobias Kildetoft Apr 13 '16 at 12:34