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.

  • $\begingroup$ 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. $\endgroup$ – Tobias Kildetoft Apr 13 '16 at 12:22
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    $\begingroup$ The word "sub". Sylow subgroup. $\endgroup$ – user1729 Apr 13 '16 at 12:23
  • $\begingroup$ 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. $\endgroup$ – almagest Apr 13 '16 at 12:25
  • $\begingroup$ @almagest, So every Sylow subgroup is a $p$-group, is that right? $\endgroup$ – MrDi Apr 13 '16 at 12:30
  • $\begingroup$ Yes, but not every $p$-subgroup is a Sylow subgroup. $\endgroup$ – 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 $C_2 \times C_4$.

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    $\begingroup$ Though any $p$-group is a Sylow subgroup of some group (e.g. itself). $\endgroup$ – Tobias Kildetoft Apr 13 '16 at 12:34

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