SylowSubgroup - Maple Help

GroupTheory

 SylowSubgroup
 construct a Sylow subgroup of a group

 Calling Sequence SylowSubgroup( p, G )

Parameters

 p - a positive rational prime G - a permutation group

Description

 • Let $G$ be a finite group, and let $p$ be a positive (rational) prime.  A Sylow $p$-subgroup of $G$ is a maximal $p$-subgroup of $G$ where, by a $p$-subgroup, we mean a subgroup whose order is a power of $p$. The Sylow theorems assert that, for a prime divisor $p$ of the order of a finite group $G$, there is a Sylow $p$-subgroup of $G$ and that all Sylow $p$-subgroups of $G$ are conjugate in $G$.  Moreover, the number of Sylow $p$-subgroups of $G$ is congruent to $1$ modulo $p$.
 • The SylowSubgroup( p, G ) command constructs a Sylow p-subgroup of a group G. The group G must be an instance of a permutation group.
 • Note that, if p is not a divisor of the order of G, then the trivial subgroup of G is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{AlternatingGroup}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{2}}{}\left({3}\right)$ (2)
 > $\mathrm{P2}≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${\mathrm{P2}}{≔}{\mathrm{}}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{P2}\right)$
 ${4}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(3,G\right)\right)$
 ${3}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5,G\right)\right)$
 ${1}$ (6)

Compatibility

 • The GroupTheory[SylowSubgroup] command was introduced in Maple 17.