Let n greaterthanorequalto 4 and we work in Sn Show that a t

Let n greaterthanorequalto 4 and we work in S_n. Show that a transposition cannot be written as a product of 3-cycles. Let sigma = (a b)(c d) be a product of two disjoint cycles (so a, b, c, d are pairwise distinct). Show that sigma is the product of two 3-cycles. Show by induction on n that if n is odd, then an n-cycle can be written as a product of 3-cycles. Show that if n is even, then an n-cycle can be written as a product of a transposition followed by 3-cycles. Show that An is generated by the 3-cycles. Show that if n greaterthanorequalto 5, then any two 3-cycles are conjugated in A_n.

Solution

Problem no.6:

Theorem: For n 5, any two 3-cycles in An are conjugate in An

Proof:

We show every 3-cycle in An is conjugate within An to (123). Let be a 3-cycle in An. It can be conjugated to (123) in Sn:

(123) = 1

for some Sn. If An we’re done.

Otherwise, let \'= (45),

so \' An and \' \'1  =(45) 1 (45)

= (45) (123) (45)

= (123).

Since An so we\'re done

Hence, For n 5, any two 3-cycles in An are conjugate in An