An Investigation into the Structure of Digroups

1. An Investigation into the Structure of Digroups Andrew Magyar, Kyle Prifogle, David White, and William Young Wabash Summer Institute in Algebra July 26, 2007

2. Introduction • Digroups and the Coquecigrue Problem • Definition of a Digroup and an Example • Trivial Digroup • Groups

3. Coquecigrue Lie Groups Linear Lie Digroups ???????? Lie Algebras Split Leibniz Algebras Leibniz Algebras Linear Lie Racks Lie Racks Racks

4. What is a Digroup? • A digroup is a set G with two binary operations,  and , a unary operation -1 and a 1 which satisfies: • G1. (G, ) and (G, ) are both semigroups (i.e.  and  are associative) • G2* x  (x  z) = (x  x)  z • G5. 1  x = x = x  1 • G6. x  x-1 = 1 = x-1 x

5. Inverses and Identities • In a digroup there can be multiple identity elements. The set of identities are defined E = {e  G| e  x = x} • The set of inverses are defined J = {x-1| x  G} • This can be defined with respect to any identity.

6. Digroup Example Identities: {1, 3, 4} 1 is a bar unit: 1-1 = 1-1 = {0, 1} 3 is not a bar unit: 3-1= {2,3} but 3-1= {3, 5}

7. Trivial Digroups • Every element is a bar unit • There is one trivial digroup of every order

8. Groups Example - 5 Note that the two operations are the same. Fact: A digroup is a group iff  =  iff the bar unit is the only identity iff |E| = 1

9. Contents • Abelian Digroups • Subdigroups • Lagrange’s Theorem • Commutant • Digroup Classification

10. Abelian Digroups A digroup is abelian iff the transpose of the  Cayley table is the  Cayley table The definition of abelian is x  y = y  x for all x, y  G

11. Subdigroups • Definition:We call a subset H of a digroup G a subdigroup if H has the structure of a digroup under the operations of G. • Theorem : In order that H be a subdigroup of digroup G, it is necessary and sufficient that (1) there exists e  H such that e is a bar-unit of H, and (2) for all f, g, l, m, n  H the elements f  e, g−1 l, and m  n−1 belong to H. • Theorem: Let G be a digroup. Consider any nonempty subset S E such that S contains a bar unit. Define H  G as H = J e 1 J e 2 J e 3 …  J e  n , where n = |S| and e i S. H is a subdigroup if J e 1 J e 2 J e 3 …  J e n = J e 1 J e 2 J e 3 …  J e n

12. Lagrange Correspondence • Lemma : J is a group • Lemma : If G is a right group, G = J  E  (J x E,  ) • Lemma: |G| = |J| * |E| • Lagrange’s Theorem for Digroups: Let G be digroup (G, , ) and H be a subdigroup of G. | H| | |G| and [J:JH] | [G: H]  |EH| | |E|.

13. Commutant • There are four potential candidates for the commutant of a digroup G • C1 = {x  G| g  x = x  g  g  G} • C2= {x  G| g  x = x  g  g  G} • C3= {x  G| g  x = x  g  g  G} • C4= {x  G| g  x = x  g  g  G}

14. Digroup Classifications There are primarily 4 types of digroups • Groups • Trivial Digroups • Principal Digroups • Permuted Digroups

15. Partitions/ Inverse Sets • For a digroup D, let |E| = m and |J| = n . The columns of the  Cayley table are broken into m partitions with n elements in each partition that are in each column m times. • Similarly the rows of the  Cayley table are broken into m partitions with n elements in each partition that are in each row m times.

16. Example D2 (6, 2) Partitions: {1,0}, {2,3}, {4,5} {1,0},{2,4},{3,5} Rows vs. Columns

17. Principal Digroups • The Principal Digroup of order n based upon a group G is notated: D 1 ( n, J ) The principal digroup is the digroup whose partitions for  are the same as those for  We say this is the Principal Digroup of order n based upon the group J.

18. Example D1 ( 6, 2) Partitions: {1,0}, {2,3}, {4,5} {1,0}, {2,3}, {4,5}

19. Arranging a Principal Digroup D1 ( 6, 2)  2

20. Therefore the number of Principle Digroups with |J|=n , |E|= m is going to be the number of ways that you can arrange J, or the number of groups of order n.

21. Therefore the number of Principle Digroups with |J|=n , |E|= m is going to be the number of ways that you can arrange J, or the number of groups of order n : • Theorem:For any group G of order n, and for any multiple mn of n, there exsists a digroup D such that the order of D is mn and the inverse set J of D is isomorphic to G. (every possible |E| has a Principal digroup) • Theorem: Two Digroups ( D, ,  ) and ( D , ,  ) with non-isomorphic J must be non-isomorphic, and with isomorphic J must be isomorphic. (there is one principal digroup for every distinct J) • Result: The number of principal digroups of order n with |E|= k is the number of groups of order n/k. • In otherwords, every non-group, non-trivial digroup has a  structure that is the same for every particular basis group J.

22. Permuted Digroups • Since all the  structures are the same for a particular J basis group all that is left is to consider digroups that have  structures with different partitions. • The Permuted Digroup of order n based upon a group J is notated: D i ( n, J ) where i > 1 The permuted digroups are the digroup whose partitions for  are different from those for .

23. Example D2 ( 6, 2) Partitions: {1,0}, {2,3}, {4,5} {1,0},{2,4},{3,5}

24. The number of Permuted Digroups • The complete structure of  is the same for all non-trivial, non-group digroups. Therefore, the permuted digroups are the only remaining possible digroups that remain to be classified. How many of them are there? Theorem: Consider a digroup ( D, ,  ) with |D|= n. The number of permuted digroups with |E| = n/p is

25. The Classification Digroups

28. Conclusion Work still to be done in the study of digroups: • The number of digroups of form D i ( n, J ) where i > 1, where |J| is composite • Sylow and Cauchy Correspondences • Class Equation? • Decomposition Theorem • The full solution to the Coquecigrue problem This research was conducted as a part of Wabash College’s Summer Institute in Algebra (WSIA) under the advisement of JD Phillips and funded by the NSF. Thank you for your time!

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