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GroupTheory

 IsMalnormal
 test whether one group is a malnormal subgroup of another

 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: $H={H}^{g}$, 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Symm}\left(3\right)$
 ${G}{≔}{{\mathbf{S}}}_{{3}}$ (1)
 > $H≔\mathrm{Subgroup}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right)\right],G\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right)⟩$ (2)
 > $\mathrm{IsMalnormal}\left(H,G\right)$
 ${\mathrm{true}}$ (3)
 > $H≔\mathrm{Subgroup}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right)\right],G\right)$
 ${H}{≔}⟨\left({1}{,}{2}{,}{3}\right)⟩$ (4)
 > $\mathrm{IsMalnormal}\left(H,G\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsMalnormal}\left(\mathrm{TrivialSubgroup}\left(G\right),G\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsMalnormal}\left(G,G\right)$
 ${\mathrm{true}}$ (8)
 > $G≔\mathrm{SmallGroup}\left(72,41\right):$
 > $\mathrm{IsFrobeniusGroup}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $H≔\mathrm{FrobeniusComplement}\left(G\right):$
 > $\mathrm{IsMalnormal}\left(H,G\right)$
 ${\mathrm{true}}$ (10)
 > $G≔\mathrm{PSL}\left(2,8\right):$
 > $S≔\mathrm{SylowSubgroup}\left(3,G\right):$
 > $\mathrm{GroupOrder}\left(S\right)$
 ${9}$ (11)
 > $\mathrm{IsCyclic}\left(S\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{IsMalnormal}\left(S,G\right)$
 ${\mathrm{true}}$ (13)
 > $G≔\mathrm{PSL}\left(2,17\right):$
 > $S≔\mathrm{SylowSubgroup}\left(3,G\right):$
 > $\mathrm{GroupOrder}\left(S\right)$
 ${9}$ (14)
 > $\mathrm{IsCyclic}\left(S\right)$
 ${\mathrm{true}}$ (15)
 > $\mathrm{IsMalnormal}\left(S,G\right)$
 ${\mathrm{true}}$ (16)

Compatibility

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