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.

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

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

$\mathrm{true}$

$G≔\mathrm{DihedralGroup}\left(4\right)\:$

$\mathrm{IsTGroup}\left(G\right)$

$\mathrm{false}$

$H≔\mathrm{Subgroup}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\right]\,G\right)$

$H≔⟨\left(1\,2\right)\left(3\,4\right)⟩$

$\mathrm{IsSubnormal}\left(H\,G\right)$

$\mathrm{true}$

$\mathrm{IsNormal}\left(H\,G\right)$

$\mathrm{false}$

$\mathrm{IsTGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{IsTGroup}\left(\mathrm{QuasicyclicGroup}\left(61\right)\right)$

$\mathrm{true}$

$\mathrm{IsTGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(5\right)\,\mathrm{CyclicGroup}\left(6\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsTGroup}\left(\mathrm{TitsGroup}\left(\right)\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

$H≔\mathrm{RandomSmallGroup}\left('\mathrm{nontrivial}'\right)\:$

$G≔\mathrm{WreathProduct}\left(H\,\mathrm{CyclicGroup}\left(2\right)\right)\:$

$\mathrm{IsTGroup}\left(G\right)$

$\mathrm{false}$

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

For more information on Maple 2024 changes, see Updates in Maple 2024.