Outer automorphism groups
The outer automorphism group of a group is the factor of the automorphism group by the inner automorphism group, the normal subgroup consisting of conjugation by group elements. A group is locally finite if every finitely generated subgroup is finite. Supposing the continuum hypothesis, we prove that there exists a locally finite p-group of continuum cardinality whose outer automorphism group is trivial. Generalizing the notion of outer automorphisms to monomorphisms, we define the outer embedding monoid of a group, which is, roughly speaking, the factor of the monoid of injective endomorphisms of the group by the inner automorphism group. We prove that a monoid is an outer embedding monoid of some group if and only if every element which is left-invertible or right-invertible is also invertible. We also prove that a monoid is an outer embedding monoid of some locally finite p-group with trivial centre if it can be written as a suitable factor of a monoid in which right cancellation holds. In particular, every group is the outer automorphism group of a locally finite p-group. We also construct a fully rigid system of such groups.