A semigroup S is defined to be a subdirect product of a non-empty collection ;We prove the class of simple groups is properly contained in the class of subdirectly irreducible groups, which is properly contained in the class of indecomposable groups. In the case of finite abelian groups, subdirect irreducibility is equivalent to indecomposability.;A Fermat group is a group whose order is a Fermat prime. The only subdirectly irreducible abelian groups with subdirectly irreducible automorphism group are ;An automorphism on a group is defined to be fixed-point-free (FPF) iff its only fixed point is the group identity element. A group K of automorphisms on a group is defined to be FPF iff each non-identity automorphism in K is FPF. For a given nontrivial odd order FPF group K of automorphisms on a group G, various sufficient conditions are specified for G and K. K is a cyclic finite direct product of subdirectly irreducible finite abelian subgroups. (Abstract shortened with permission of author.)
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