Prove that A n is simple for n 5, following the steps given. a. Show A
Question:
Prove that An is simple for n ≥ 5, following the steps given.
a. Show An contains every 3-cycle if n ≥ 3.
b. Show An is generated by the 3-cycles for n ≥ 3.
c. Let r and s be fixed elements of {l, 2, • • • , n} for n ≥ 3. Show that An is generated by the n "special" 3-cycles of the form (r, s, i) for 1 ≤ i ≤ n.
d. Let N be a normal subgroup of An for n ≥ 3. Show that if N contains a 3-cycle, then N = An
e. Let N be a nontrivial normal subgroup of An for n ≥ 5. Show that one of the following cases must hold, and conclude in each case that N = An.
Case I N contains a 3-cycle.
Case II N contains a product of disjoint cycles, at least one of which has length greater than 3.
Case III N contains a disjoint product of the form σ = µ,(a4 , a5, a6)(a1, a2, a3).
Case IV N contains a disjoint product of the form σ = µ,(a1, a2, a3) where µ, is a product of disjoint2-cycles.
Case V N contains a disjoint product a of the form σ = µ,(a3, a4)(a1, a2), whereµ, is a product of an even number of disjoint 2-cycles.
Step by Step Answer:
a Because a b c a ca b we see that every 3cycle is an even permutation and hence is in A n b Let An and write as a product of transpositions The number of transpositions in the product will be even by ...View the full answer
