Perfect group

Definition
A group is said to be perfect if it satisfies the following equivalent conditions:

In terms of the fixed-point operator
The property of being perfect is obtained by applying the fixed-point operator to a subgroup-defining function, namely the derived subgroup.

Extreme examples

 * The trivial group is a perfect group.

Groups dissatisfying the property
Note that any nontrivial solvable group cannot be a perfect group, so this gives lots of non-examples. The examples discussed below concentrate more on the non-solvable groups that still fail to be perfect.

Testing
To test whether a given group is perfect, the command is:

IsPerfectGroup(group);

The command:

PerfectGroup(n,r)

gives the $$r^{th}$$ perfect group of order $$n$$. If $$r$$ is not specified, this simply gives the first perfect group of order $$n$$. An error is thrown if there are no perfect groups of order $$n$$.