The discussion revolves around methods for proving that a particular subset of the symmetric group S4 is a normal subgroup. Participants explore various approaches, including the implications of Lagrange's theorem and the representation of permutations as products of disjoint transpositions. The focus is on finding alternative methods to brute force verification.
There is no consensus on the effectiveness of the proposed methods, as some participants express uncertainty and seek clarification on the application of Lagrange's theorem and the utility of transpositions.
Participants have not fully explored the implications of Lagrange's theorem or the method of using transpositions, leaving some assumptions and steps unresolved.
tauon said:you can use a certain (obvious) implication of the Lagrange theorem for groups to thin out the workload; and you can also use the fact that any permutation is a product of disjoint transpositions to further simplify notation... of course if by S4 you mean S_4 the symmetry group.
so yes there is a way to find the normal subgroups by other means than brute force.
tauon said:you can use a certain (obvious) implication of the Lagrange theorem for groups to thin out the workload; and you can also use the fact that any permutation is a product of disjoint transpositions to further simplify notation... of course if by S4 you mean S_4 the symmetry group.
so yes there is a way to find the normal subgroups by other means than brute force.