The influence of the existence of a group automorphism of a certain kind is the subject of much research. This paper will attempt to summarize one small corner of this research. We survey automorphisms that map most of the elements of a finite group to powers of themselves and the groups that support such automorphisms. The types of groups which have such automorphisms is limited in some cases, and in such cases, the proportion of elements taken to their powers can have only certain values.

The following notation will be used in this paper.

• Z(G): The centre of the group G.

• G′: The commutator subgroup of G.

• Gp: The group generated by all of the pth powers of elements in G.

• (g)φ: The image of the group element g under the automorphism φ.

• |G : A|: The index of the subgroup A in the group G.

• |T|: The number of elements in the set T.

• Tn(φ) : {g ∈ G : (g)φ = gn} for an automorphism φ of G.

• Lp: The set of all finite groups with order divisible by a prime p, but by no smaller prime.

It is well-known that if the map φ : x → x2 for all x ∈ G is an automorphism of the group G, then G is abelian and has odd order if it is finite. The existence of automorphisms of a group G of the form φ : x → x-1 or φ : x → x3 where G is finite also force G to be abelian.