The semigroups of order 9 and their automorphism groups

Andreas Distler, Tom Kelsey

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52 989 400 714 478 and up to isomorphism to be 105 978 177 936 292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148 195 347 518 186 distinct associative binary operations on an 8-element set and 38 447 365 355 811 944 462 on a 9-element set.
Original languageEnglish
Pages (from-to)93-112
JournalSemigroup Forum
Issue number1
Early online date21 Jun 2013
Publication statusPublished - Feb 2014


  • Semigroup
  • Automorphism group
  • Enumeration


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