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GroupTheory

  

IsPermutable

  

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

 

Calling Sequence

Parameters

Description

Examples

Compatibility

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 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.

Examples

withGroupTheory:

GGroupPerm1,2,3,6,4,5,7,8,Perm2,5,6,8

G1,2,3,6,4,5,7,8,2,56,8

(1)

GroupOrderG

16

(2)

HSubgroupPerm2,5,6,8,G

H2,56,8

(3)

IsPermutableH,G

true

(4)

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

IsNormalH,G

false

(5)

Permutable subgroups are subnormal.

IsSubnormalH,G

true

(6)

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

andmapIsPermutable,NormalSubgroupsG,G

true

(7)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[IsNormal]

GroupTheory[IsSubgroup]

GroupTheory[IsSubnormal]