Powers of Cycle-Classes in Symmetric Groups

✍ Scribed by Edward Bertram; Marcel Herzog

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 136 KB
• Volume: 94
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3131

✦ Synopsis

Theorem 1 [Be, Corollary 2.1]. Each permutation in the alternating group A n , n 2, is a product of two l-cycles in S n if and only if either w 3n 4 x l n or n=4 and l=2.

Note. Theorem 1 implies that when n=4 every even permutation is a product of two l-cycles if and only if 2 l 4.

The aim of the present paper is to extend this result to three and four l-cycles. Our main results are Theorems 2 and 3: Theorem 2. Each permutation in A n , n 1, is a product of three l-cycles in S n if and only if l is odd and either W n 2 X l n or n=7 and l=3.

Theorem 3. Each permutation in A n , n 2, is a product of four l-cycles in S n if and only if :

(1) W 3n 8 X l n if n 1 (mod 8); (2) w 3n 8 x l n if n#1, 0 (mod 8); (3) n=6 and l=2.