The subgroups of the symmetric group S4

Description

The program in GAP lists each subgroup of $S_4$, the group of the permutations of $\{1, 2, 3, 4\}$. It writes each subgroup with the sorted list of its elements to the certificate, and it prints the number of subgroups of each order.

The checker is a short Python program that does not use GAP. It finds the subgroups of $S_4$ again with its own search. It starts with the trivial subgroup and adds the subgroup that a known subgroup and one more element generate, until it finds no new subgroup. It accepts the certificate only when the lines of the certificate are these subgroups, each one time, and the order of each one divides 24.

Remarks

The orders divide 24, as Lagrange's theorem says. The converse of the theorem is not true for each finite group, but it is true for $S_4$. For each divisor $d$ of 24, $S_4$ has a subgroup of order $d$. The alternating group $A_4$ of order 12 has no subgroup of order 6, so the converse is false for $A_4$.

m2 compute verify AAYEQE