GroupTheory/IsSubnormal - Maple Help

GroupTheory

 IsSubnormal
 test whether one group is contained as a subnormal subgroup of another

 Calling Sequence IsSubnormal( H, G )

Parameters

 H - a group G - a group

Description

 • A group $H$ is a subnormal  subgroup of a group $G$ if $H$ is a subgroup of $G$, and if there is a chain

$G={G}_{0}▹{G}_{1}▹\dots ▹H$

 such that ${G}_{k}$ is normal in ${G}_{k-1}$, for each $i$. Every normal subgroup of a group is subnormal, but not conversely.
 • The IsSubnormal( H, G ) command tests whether the group H is a subnormal subgroup of the group G.  It returns true if H is subnormal in G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsSubnormal cannot determine whether H is a subnormal subgroup of G.

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)
 > $\mathrm{IsSubnormal}\left(H,G\right)$
 ${\mathrm{true}}$ (4)

Every normal subgroup of a group is subnormal.

 > $\mathrm{andmap}\left(\mathrm{IsSubnormal},\mathrm{NormalSubgroups}\left(G\right),G\right)$
 ${\mathrm{true}}$ (5)

Compatibility

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