Alternating Groups #
The alternating group on a finite type
α is the subgroup of the permutation group
consisting of the even permutations.
Main definitions #
Main results #
two_mul_card_alternating_groupshows that the alternating group is half as large as the permutation group it is a subgroup of.
closure_three_cycles_eq_alternatingshows that the alternating group is generated by 3-cycles.
alternating group permutation
A key lemma to prove $A_5$ is simple. Shows that any normal subgroup of an alternating group on at least 5 elements is the entire alternating group if it contains a 3-cycle.
Part of proving $A_5$ is simple. Shows that the square of any element of $A_5$ with a 3-cycle in its cycle decomposition is a 3-cycle, so the normal closure of the original element must be $A_5$.
The normal closure of $(04)(13)$ within $A_5$ is the whole group. This will be used to show that the normal closure of any permutation of cycle type $(2,2)$ is the whole group.
Shows that any non-identity element of $A_5$ whose cycle decomposition consists only of swaps is conjugate to $(04)(13)$. This is used to show that the normal closure of such a permutation in $A_5$ is $A_5$.