On Algebras with many symmetric operations
  We show that, for each finite algebra A, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by A has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point- free automorphism. 
Catarina Carvalho and Andrei Krokhin
math.RA