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GroupTheory

  

IsMalnormal

  

test whether one group is a malnormal subgroup of another

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsMalnormal( H, G )

Parameters

H

-

a permutation group

G

-

a permutation group

Description

• 

A group H is a malnormal subgroup of a group G if H is a subgroup of G, and if it is has trivial intersection with each of its conjugates: H=Hg, for all g in G.

• 

For example, any non-normal subgroup of prime order is malnormal. On the other hand, a cyclic Sylow subgroup of a finite simple group is malnormal, even if its order is not prime.

• 

A group is Frobenius if, and only if, it possesses a non-trivial, proper malnormal subgroup, the Frobenius complement.

• 

The IsMalnormal( H, G ) command tests whether the group H is a malnormal subgroup of the group G.  It returns true if H is malnormal in G, and returns false otherwise.  For some pairs H and G of groups, the value FAIL may be returned if IsMalnormal cannot determine whether H is a malnormal subgroup of G.

Examples

withGroupTheory:

GSymm3

GS3

(1)

HSubgroupPerm1,2,G

H1,2

(2)

IsMalnormalH,G

true

(3)

HSubgroupPerm1,2,3,G

H1,2,3

(4)

IsMalnormalH,G

false

(5)

IsNormalH,G

true

(6)

IsMalnormalTrivialSubgroupG,G

true

(7)

IsMalnormalG,G

true

(8)

GSmallGroup72,41:

IsFrobeniusGroupG

true

(9)

HFrobeniusComplementG:

IsMalnormalH,G

true

(10)

GPSL2,8:

SSylowSubgroup3,G:

GroupOrderS

9

(11)

IsCyclicS

true

(12)

IsMalnormalS,G

true

(13)

GPSL2,17:

SSylowSubgroup3,G:

GroupOrderS

9

(14)

IsCyclicS

true

(15)

IsMalnormalS,G

true

(16)

Compatibility

• 

The GroupTheory[IsMalnormal] command was introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

GroupTheory

GroupTheory[FrobeniusComplement]

GroupTheory[IsFrobenius]

GroupTheory[IsNormal]

GroupTheory[IsSubgroup]

GroupTheory[SymmetricGroup]