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.

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

$G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1\,2\,3\,6\,4\,5\,7\,8\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,5\right]\,\left[6\,8\right]\right]\right)\right)$

$G≔⟨\left(1\,2\,3\,6\,4\,5\,7\,8\right)\,\left(2\,5\right)\left(6\,8\right)⟩$

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

$16$

$H≔\mathrm{Subgroup}\left(\left\{\mathrm{Perm}\left(\left[\left[2\,5\right]\,\left[6\,8\right]\right]\right)\right\}\,G\right)$

$H≔⟨\left(2\,5\right)\left(6\,8\right)⟩$

$\mathrm{IsPermutable}\left(H\,G\right)$

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

For more information on Maple 2018 changes, see Updates in Maple 2018.