GroupTheory/IsTGroup - Maple Help

GroupTheory

 IsTGroup
 determine whether a group is a T-group

 Calling Sequence IsTGroup( G )

Parameters

 G - a permutation group

Description

 • A group $G$ is said to be a T-group if every subnormal subgroup of $G$ is normal in $G$. This is equivalent to the assertion that normality is a transitive relation on the subgroup lattice of $G$.
 • Every abelian group is a T-group, as is every simple group.
 • The IsTGroup( G ) command determines whether the group G is a T-group. It returns the value true if G is a T-group, and returns false otherwise.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

The smallest non-abelian T-group is the symmetric group of degree $3$.

 > $\mathrm{IsTGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (1)

The smallest group that is not a T-group is the dihedral group of order $8$.

 > $G≔\mathrm{DihedralGroup}\left(4\right):$
 > $\mathrm{IsTGroup}\left(G\right)$
 ${\mathrm{false}}$ (2)

The following subgroup is subnormal but not normal in G.

 > $H≔\mathrm{Subgroup}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)\right],G\right)$
 ${H}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩$ (3)
 > $\mathrm{IsSubnormal}\left(H,G\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsNormal}\left(H,G\right)$
 ${\mathrm{false}}$ (5)

It is the only group of order $8$ that is not a T-group, since the other non-abelian group of that order is the quaternion group of order $8$, which is Hamiltonian, and hence, is also a T-group.

 > $\mathrm{IsTGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (6)

The alternating group of degree $4$ is the only alternating group that is not a T-group. Since the alternating group of degree $5$, being simple, is a T-group, this shows that subgroups of T-groups need not be T-groups. On the other hand, subgroups of soluble T-groups are T-groups.

 > $\mathrm{IsTGroup}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{false}}$ (7)

All abelian groups are T-groups.

 > $\mathrm{IsTGroup}\left(\mathrm{QuasicyclicGroup}\left(61\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsTGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(5\right),\mathrm{CyclicGroup}\left(6\right)\right)\right)$
 ${\mathrm{true}}$ (9)

Every simple group is a T-group.

 > $\mathrm{IsTGroup}\left(\mathrm{TitsGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (10)

An example of an insoluble T-group that is not simple.

 > $\mathrm{IsTGroup}\left(\mathrm{SL}\left(2,7\right)\right)$
 ${\mathrm{true}}$ (11)

An example of a soluble group that is not a T-group.

 > $\mathrm{IsTGroup}\left(\mathrm{SL}\left(2,3\right)\right)$
 ${\mathrm{false}}$ (12)

If $H$ is any non-trivial group, then the wreath product of $H$ with a cyclic group of order $2$ is not a T-group.

 > $H≔\mathrm{RandomSmallGroup}\left('\mathrm{nontrivial}'\right):$
 > $G≔\mathrm{WreathProduct}\left(H,\mathrm{CyclicGroup}\left(2\right)\right):$
 > $\mathrm{IsTGroup}\left(G\right)$
 ${\mathrm{false}}$ (13)

Compatibility

 • The GroupTheory[IsTGroup] command was introduced in Maple 2024.