On the Efficiency of Finite Simple Semigroups
Hayrullah Ayık, Colin Matthew Campbell, John J O'Connor, Nik Ruškuc · DergiPark (Istanbul University) · 2000
Let S be a finite simple semigroup, given as a Rees matrix semigroup \\mathcal{M}[G;I,L ;P] over a group G. We prove that the second homology of S is H2(S)=H2(G)\\times {\\mathbb Z}(|I|-1)(|L |-1). It is known that for any finite presentation \\langle \\: A\\: |\\: R\\: \\rangle of S we have |R|-|A|\\geq \\mbox{rank}(H2(S)); we say that S is efficient if equality is attained for some presentation. Given a presentation \\langle \\: A1\\: |\\: R1\\: \\rangle for G, we find a presentation \\langle \\: A\\: |\\: R\\: \\rangle for S such that |R|-|A|=|R1|-|A1|+(|I|-1)(|L |-1)+1. Further, if R1 contains a relation of a special form, we show that |R|-|A| can be reduced by one. We use this result to prove that S is efficient whenever G is finite abelian or dihedral of even degree.