# Conjugacy of pointwise conjugate homomorphisms

Von | Oktober 7, 2020

Given groups and , let us say that the pair has property (*) if any two injective homomorphisms are conjugate if (and only if) they are pointwise conjugate.

Problem 1: Describe the class of groups such that has (*) for every group .

Problem 2: Describe the class of groups such that has (*) for every group .

Definitions: Given a group , two elements are conjugate if there exists such that . Two homomorphisms are pointwise conjugate if is conjugate to in for every . (Thus, for each there exists such that .) Further, and are conjugate if there exists such that for every . If and are conjugate, then they are locally conjugate. Property (*) records that the converse holds (for injective homomorphisms).

Motivation: Problem 30 of the Scottish Book ​​ (which is still open) asks to determine which groups have the following property: Pairs and in are conjugate (there exists such that and ) if (and only if) for every word in two noncommuting variables the elements and are conjugate. Using property (*) formulated above, Problem 30 asks to determine which groups have the property that satisfies (*) for every subgroup of that is generated by two elements. The above Problem 2 is a more general (and possibly more natural) version of this problem.

An automorphism is class-preserving if is conjuagte to for every . If has (*), then every class-preserving automorphism of is inner. The study of groups with (or without) outer class-preserving automorphisms has a long history; see for instance ​​ and ​​. There exist finite groups with outer class-preserving automorphisms. In particular, there exist finite groups that belong neither to nor to . Note also that contains all abelian groups and that contains all cyclic groups.

1. 
R.D. Mauldin, The Scottish Book, Springer International Publishing, 2015. https://doi.org/10.1007/978-3-319-22897-6.
2. 
C.-H. Sah, Automorphisms of finite groups, Journal of Algebra. (1968) 47–68. https://doi.org/10.1016/0021-8693(68)90104-x.
3. 
M.K. Yadav, Class preserving automorphisms of finite p-groups: a survey, in: C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal, G.C. Smith, G. Traustason (Eds.), Groups St Andrews 2009 in Bath, Cambridge University Press, 2007: pp. 569–579. https://doi.org/10.1017/cbo9780511842474.019.