Normal Subgroups

James F. Hurley

Maple's group package can determine whether a subgroup H of a group G is normal. After invoking the group package, you need to specify both G and H . You then can ask Maple whether H "i snormal " in G .

As in the
Groups and Cosets worksheet, the second command below defines the symmetric group of degree 3 as the group generated by the transposition g = ( 2 3 ) and the 3-cycle f = ( 1 2 3 ). The next command defines the subgroup H to be the cyclic group generated by f . Since we know that H is a subgroup of order 3 in a group of order 6, it has index 2 and so is indeed a normal subgroup. The last line shows the syntax for the isnormal command. Execute the routine to see Maple's answer to the question "is H normal in G "? [Recall that to execute the commands, position the cursor at the end after the last semicolon and hit the Enter key (or press the Shift and Return keys simultaneously).]

> with(group):
S_3 := permgroup( 3, {[[2, 3]], [[1, 2, 3]]} );
H := permgroup( 3, {[[1, 2, 3]]} );
isnormal(S_3, H);

Could it be that the cyclic subgroup H generated by the 3-cycle ( 1 2 3 ) is always normal in , no matter the value of n ? To investigate, let's ask Maple whether the subgroup generated in by that 3-cycle ( 1 2 3) is normal. Note that is generated by all transpositions ( i j ), of which there are exactly 6 = (4*3)/2.

> G := permgroup( 4, {[[1, 2]], [[1, 3]], [[1, 4]], [[2, 3]], [[2, 4]], [[3, 4]]} );
H := permgroup( 4, {[[1, 2, 3]]} );
isnormal(G, H);