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A180423 Number of unbreakable loops of order n. 0
2, 28, 9906, 43803136 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=5..8.

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


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


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

Sequence in context: A058502 A080266 A308757 * A090497 A128371 A175932

Adjacent sequences: A180420 A180421 A180422 * A180424 A180425 A180426




Jonathan Vos Post, Sep 03 2010



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