Subgroups are studied in most first courses in abstract algebra. Computing subgroups of a cyclic group is pretty straight forward, but it’s harder to get students to understand subgroups of a non-cyclic or, even better, a non-abelian group. With *Magma*, it’s easy to compute examples. Let’s start with something simple.

` > A5 := AlternatingGroup(5); `

` > A5; `

` Permutation group A5 acting on a set of cardinality 5 `

` Order = 60 = 2^2 * 3 * 5 `

` (3, 4, 5) `

` (1, 2, 3) `

Computing the subgroups of A5 is easy in *Magma*. Trying executing the following command and see if you can understand what it is giving you:

` > Subgroups(A5); `

We can do the same for cyclic subgroups:

` > CSubs := CyclicSubgroups(A5); `

` > CSubs; `

Or abelian subgroups:

` > ASubs := AbelianSubgroups(A5); `

` > ASubs; `

*Magma* will allow you to access (abstractly) one of the subgroups by choosing its location in the list followed by “`subgroup”:

` > S := ASubs[5]`subgroup; `

` > S; `

` Permutation group S acting on a set of cardinality 5 `

` Order = 4 = 2^2 `

` (2, 5)(3, 4) `

` (2, 4)(3, 5) `

** Exercise:** Create the dihedral group acting on an octagon. Find all the cyclic subgroups, and give a geometric description of those with order 2. Start by brainstorming for the answer and then use

*Magma*to confirm your beliefs. You can guess the command for generating the Dihedral Group acting on a 8-sided figure, can’t you?