A180423 Number of unbreakable loops of order n.

2, 28, 9906, 43803136

Offset: 5

Author

Jonathan Vos Post, Sep 03 2010

Keywords

• nonn

Comments

From p. 3 of Beaudry, Figure 1: Unbreakable loops of size 5 to 9. We say that a finite loop is unbreakable whenever it doesn't have proper subloops, that is, other than itself and the trivial one-element loop. While it is easy to see that the finite associative unbreakable loops are exactly the cyclic groups of prime order, it turns out that finite, nonassociative unbreakable loops are numerous and diverse. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We describe two families of commutative unbreakable loops of odd order, n >= 7, one where the loop's multiplication group is isomorphic to the alternating group A_n and another where the multiplication group is isomorphic to the symmetric group S_n. We also prove for each even n >= 6 that there exist unbreakable loops of order n whose multiplication group is isomorphic to S_n.

Examples

			a(5) = 2 because there are 6 loops of order 5, of which 2 are unbreakable.
		

Links

• Martin Beaudry, Louis Marchand, Unbreakable Loops, Sep 02, 2010.

Crossrefs

Cf. A057771 Number of loops (quasigroups with an identity element) of order n.