You can test whether a subgroup of a finite group is subnormal or permutable (quasi-normal) by using the new IsSubnormal and IsPermutable commands. These are generalizations of normal subgroups.

with( GroupTheory ):

G := Symm( 4 );

$G≔{\mathbf{S}}_4$

H := Subgroup( { Perm( [[1,2],[3,4]] ) }, G );

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

IsSubnormal( H, G );

$\mathrm{true}$

IsPermutable( H, G );

$\mathrm{false}$

IsNormal( H, G );

$\mathrm{false}$

G := Symm( 3 );

$G≔{\mathbf{S}}_3$

N := Subgroup( { Perm( [[1,2,3]] ) }, G );

$N≔⟨\left(1\,2\,3\right)⟩$

IsNormal( N, G );

$\mathrm{true}$

IsPermutable( N, G );

$\mathrm{true}$

IsSubnormal( N, G );

$\mathrm{true}$

To check whether a finite group is a $p$-group, for a prime number $p$, use the new IsPGroup command. The PGroupPrime command can be used to return the prime number $p$ for which a group is a finite $p$-group.

IsPGroup( Symm( 3 ) );

$\mathrm{false}$

IsPGroup( DihedralGroup( 4 ) );

$\mathrm{true}$

PGroupPrime( DihedralGroup( 4 ) );

$2$

IsPGroup( DirectProduct( CyclicGroup( 128 ), QuaternionGroup() ) );

$\mathrm{true}$

The ElementOrder command has been extended to work on elements of finite finitely presented groups.

G := < a, b | a^2, b^3, (a.b)^5 = 1 >:

ElementOrder( a.b^2 .a, G );

$3$

The new ElementPower command computes powers of elements of a permutation group or a Cayley table group.

The new ClassNumber command returns the number of conjugacy classes of a finite group. In some cases, this can be faster than actually computing the conjugacy classes themselves and counting them.

ClassNumber( Symm( 3 ) );

$3$

ClassNumber( DirectProduct( Monster(), DihedralGroup( 4 * n ) ) ) assuming n :: posint;

$388n+582$

The number of Abelian groups of a given order can be computed by using the new NumAbelianGroups command.

NumAbelianGroups( 1000 );

$9$