test whether one group is contained as a permutable subgroup of another
IsPermutable( H, G )
IsQuasinormal( H, G )
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 KH=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.
G ≔ Group⁡Perm⁡1,2,3,6,4,5,7,8,Perm⁡2,5,6,8
H ≔ Subgroup⁡Perm⁡2,5,6,8,G
This is the smallest example of a group with a permutable, non-normal subgroup.
Permutable subgroups are subnormal.
Of course, all the normal subgroups of a group are permutable.
The GroupTheory[IsPermutable] command was introduced in Maple 2018.
For more information on Maple 2018 changes, see Updates in Maple 2018.
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