One of the most powerful uses of *Magma* is with group theory. We will only be able to touch the surface of what *Magma* can do, but this should give you enough to see the power. Let’s start simply. We’ll define the symmetric group on 4 elements and pull out an element to examine.

`> S4 := SymmetricGroup(4);`

> S4;

Symmetric group S4 acting on a set of cardinality 4

Order = 24 = 2^3 * 3

> p := S4!(1,2,3);

> Order(p);

3

It’s easy to create subgroups generated by elements.

`> S := sub<S4 | p>;`

> S;

Permutation group S acting on a set of cardinality 4

Order = 3

(1, 2, 3)

We can also create permutation groups by giving a list of generators. Let’s look at the Klein 4 group and show that it is not cyclic.

`Klein4 := PermutationGroup<4 | (1,2)(3,4), (1,3)(2,4) >;`

> #Klein4;

4

> IsAbelian(Klein4);

true

> IsCyclic(Klein4);

false

> C4 := CyclicGroup(4);

> Klein4,C4;

Permutation group Klein4 acting on a set of cardinality 4

Order = 4 = 2^2

(1, 2)(3, 4)

(1, 3)(2, 4)

Permutation group C4 acting on a set of cardinality 4

Order = 4 = 2^2

(1, 2, 3, 4)

> IsIsomorphic(Klein4,C4);

false

** Exercise:** Since

*Magma*was written by mathematicians for mathematicians, it’s often easy to guess the commands for certain things. Try to do the following without looking up the commands:

- Use
*Magma*to show that the symmetric group acting on a set of cardinality 3 is the same as the dihedral group acting on a set of cardinality 3. - Construct a permutation group acting on 4 elements generated by the cycle elements (1,2)(3,4) and (1,2,3). Check if this group is the same as the alternating group A
_{4}.