 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 or Cayley table 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 or a Cayley table 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)
 > $\mathrm{P2}≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${\mathrm{P2}}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩$ (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)
 > $G≔\mathrm{CayleyTableGroup}\left(\mathrm{Symm}\left(5\right)\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 120 elements >}}$ (7)
 > $P≔\mathrm{SylowSubgroup}\left(3,G\right)$
 ${P}{≔}{\mathrm{< a Cayley table group with 3 elements >}}$ (8)
 > $N≔\mathrm{Normaliser}\left(P,G\right)$
 ${N}{≔}{{N}}_{{\mathrm{< a Cayley table group with 120 elements >}}}{}\left({\mathrm{< a Cayley table group with 3 elements >}}\right)$ (9)
 > $Q≔\mathrm{SylowSubgroup}\left(2,N\right)$
 ${Q}{≔}{\mathrm{< a Cayley table group with 4 elements >}}$ (10)
 > $\mathrm{GroupOrder}\left(Q\right)$
 ${4}$ (11) Compatibility

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