IsPermutable - Maple Help

GroupTheory

 IsPermutable
 test whether one group is contained as a permutable subgroup of another

 Calling Sequence IsPermutable( H, G ) IsQuasinormal( H, G )

Parameters

 H - a group G - a group

Description

 • A group $H$ is a permutable (or quasi-normal) subgroup of a group $G$ if $H$ is a subgroup of $G$, and if it permutes (set-wise) with every other subgroup $K$ of $G$ in the sense that $\mathrm{KH}=\mathrm{HK}$. Every normal subgroup of a group is permutable, but not conversely.
 • The IsPermutable( H, G ) command tests whether the group H is a permutable subgroup of the group G.  It returns true if H is permutable in G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsPermutable cannot determine whether H is a permutable subgroup of G.
 • The IsQuasinormal command is an alias for IsPermutable.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,2,3,6,4,5,7,8\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,5\right],\left[6,8\right]\right]\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}{,}{6}{,}{4}{,}{5}{,}{7}{,}{8}\right){,}\left({2}{,}{5}\right)\left({6}{,}{8}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${16}$ (2)
 > $H≔\mathrm{Subgroup}\left(\left\{\mathrm{Perm}\left(\left[\left[2,5\right],\left[6,8\right]\right]\right)\right\},G\right)$
 ${H}{≔}⟨\left({2}{,}{5}\right)\left({6}{,}{8}\right)⟩$ (3)
 > $\mathrm{IsPermutable}\left(H,G\right)$
 ${\mathrm{true}}$ (4)

This is the smallest example of a group with a permutable, non-normal subgroup.

 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{false}}$ (5)

Permutable subgroups are subnormal.

 > $\mathrm{IsSubnormal}\left(H,G\right)$
 ${\mathrm{true}}$ (6)

Of course, all the normal subgroups of a group are permutable.

 > $\mathrm{andmap}\left(\mathrm{IsPermutable},\mathrm{NormalSubgroups}\left(G\right),G\right)$
 ${\mathrm{true}}$ (7)

Compatibility

 • The GroupTheory[IsPermutable] command was introduced in Maple 2018.