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 [1] (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 [2] and [3]. 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.