A226142 The smallest positive integer k such that the symmetric group S_n is a product of k cyclic groups.

1, 1, 2, 3, 3, 4, 4

Offset: 1

Author

W. Edwin Clark, May 27 2013

Keywords

• nonn
• hard
• more
• nice

Comments

Since S_{n+1} is a product of a subgroup isomorphic to S_n and the cyclic group <(1,2,3,...,n+1)> we have a(n+1) <= a(n) + 1. On the other hand it is not clear that a(n) <= a(n+1) for all n. A lower bound is given by A226143(n) = ceiling(log(m)(n!)), m = A000793(n), a sequence that is not nondecreasing.

This sequence was suggested by a posting of L. Edson Jeffery on the seqfans mailing list on May 24, 2013.

Cardinality of the smallest subset(s) X of S_n such that every permutation in S_n can be expressed as a product of some elements in X. - Joerg Arndt, Dec 13 2015

Examples

			a(7) = 4 since a factorization of S_7 is given by C_1*C_2*C_3*C_4 where
C_1 = <(1,2,3,4)(5,6,7)>,
C_2 = <(1,4,6)(2,3,5,7)>,
C_3 = <(1,2,5,7)(3,4,6)>,
C_4 = <(1,3,5,6,7)(2,4)>,
and a brute force computation shows that S_7 is not a product of 3 or fewer cyclic subgroups.
		

Links

• Miklós Abért, Symmetric groups as products of Abelian subgroups, Bull. Lond. Math. Soc., Volume 34, Issue 04, July 2002, pp. 451-456.

Crossrefs

Cf. A051625, A225788.