Subgroups

The .subgroup() method in SageMath produces the cyclic subgroup generated by the set of generators provided to the method. That is \[ G.\text{subgroup}([a,b,c]) = \langle a^ib^jc^k\mid i,j,k\in \mathbb{N}\cup \{0\}\rangle \] This returns the subgroup generated by $a,b$ and $c$, which we can then work with just like a normal group in SageMath. An example of this follows.

A note that should be made about Cayley Tables, they don't often have the information that we want on them, that is, the names that SageMath gives to the elements of the group may not be intuitive. We can get around that by having SageMath change the labels on the Cayley Table. We go through this in the following cell.

We have another example of doing this in the following cell.