Outer automorphisms of locally finite \(p\)-groups.
Let \(G\) be any group. Recall that the set \(\text{Inn}(G)\) of all inner automorphisms of \(G\) is a normal subgroup of \(\Aut(G)\). The quotient group \(\Aut(G)/\text{Inn}(G)\) is called the group of all outer automorphisms of \(G\) and is denoted by \(\text{Out}(G)\). Several authors have studied outer automorphism groups in the literature. For instance, Matumoto showed that any group is the outer automorphism group of some group. In fact, Paras and the second author proved that the same result holds if we restrict ourselves to the class of all metabelian groups. This paper is devoted to the question of what groups may appear as the outer automorphism group of a locally finite \(p\)-group. Here a \(p\)-group is locally finite if every finitely generated subgroup is (nilpotent) finite. An old result by Gaschütz shows that any finite \(p\)-group with at least 3 elements has a non-trivial outer automorphism and the same holds for infinite nilpotent \(p\)-groups. However, answering a problem of Hall, the second author and Droste proved that every countable locally finite \(p\)-group is the outer automorphism group of some locally finite \(p\)-group. The authors of this nice paper strengthen this result substantially by showing that every group is the outer automorphism group of some locally finite \(p\)-group. The construction uses the existence of complicated Abelian \(p\)-groups which follows from the application of Shelah's Black Box. The paper is easy to read and every step of the construction is explained clearly. It is therefore an interesting contribution to the theory of groups and recommended to every mathematician interested in it.
scientific article; zbMATH DE number 1955947
scientific article