Unary operations with long pre-periods

It is well known that the congruence lattice ConAConA of an algebra AA is uniquely determined by the unary polynomial operations of AA (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let AA be a finite algebra with |A|=n|A|=n. If Imf=AImf=A or |Imf|=1|Imf|=1 for every unary polynomial operation f   of AA, then AA is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f:A→Af:A→A is not a permutation then A⊃ImfA⊃Imf and there is a least natural number λ(f)λ(f) with Imfλ(f)=Imfλ(f)+1Imfλ(f)=Imfλ(f)+1. We consider unary operations with λ(f)=n-1λ(f)=n-1 for n⩾2n⩾2 and λ(f)=n-2λ(f)=n-2 for n⩾3n⩾3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.