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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SL}\left(2\,5\right)\:$

$\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$

${\left(2\right)}^3\left(3\right)\left(5\right)$

$\mathrm{P2}≔\mathrm{SylowSubgroup}\left(2\,G\right)$

$\mathrm{P2}≔⟨\left(1\,2\,3\,4\right)\left(5\,20\,15\,10\right)\left(6\,22\,18\,14\right)\left(7\,23\,19\,11\right)\left(8\,24\,16\,12\right)\left(9\,21\,17\,13\right)\,\left(1\,5\,3\,15\right)\left(2\,10\,4\,20\right)\left(6\,8\,18\,16\right)\left(7\,13\,19\,21\right)\left(9\,23\,17\,11\right)\left(12\,14\,24\,22\right)⟩$

$\mathrm{GroupOrder}\left(\mathrm{P2}\right)$

$8$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(3\,G\right)\right)$

$3$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5\,G\right)\right)$

$5$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(7\,G\right)\right)$

$1$

$G≔\mathrm{CayleyTableGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$G≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 24\; elements\; >}$

$P≔\mathrm{SylowSubgroup}\left(3\&comma;G\right)$

$P≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 3\; elements\; >}$

$N≔\mathrm{Normaliser}\left(P\&comma;G\right)$

$N≔N_{\mathrm{<\; a\; Cayley\; table\; group\; with\; 24\; elements\; >}}\left(\mathrm{<\; a\; Cayley\; table\; group\; with\; 3\; elements\; >}\right)$

$Q≔\mathrm{SylowSubgroup}\left(2\&comma;N\right)$

$Q≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 2\; elements\; >}$

$\mathrm{GroupOrder}\left(Q\right)$

$2$

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

For more information on Maple 17 changes, see Updates in Maple 17.