Conjugacy of pointwise conjugate homomorphisms

Von | Oktober 7, 2020

Given groups H and G, let us say that the pair (H,G) has property (*) if any two injective homomorphisms \alpha_0,\alpha_1\colon H\to G are conjugate if (and only if) they are pointwise conjugate.

Problem 1: Describe the class C_1 of groups H such that (H,G) has (*) for every group G.

Problem 2: Describe the class C_2 of groups G such that (H,G) has (*) for every group H.

Definitions: Given a group G, two elements a,b\in G are conjugate if there exists x\in G such that a=xbx^{-1}. Two homomorphisms \alpha_0,\alpha_1\colon H\to G are pointwise conjugate if \alpha_0(a) is conjugate to \alpha_1(a) in G for every a\in H. (Thus, for each a\in H there exists x_a\in G such that \alpha_0(a)=x_a\alpha_1(a)x_a^{-1}.) Further, \alpha_0 and \alpha_1 are conjugate if there exists x\in G such that \alpha_0(a)=x\alpha_1(a)x^{-1} for every a\in H. If \alpha_0 and \alpha_1 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 G have the following property: Pairs (a,a') and (b,b') in G are conjugate (there exists x\in G such that a=xbx^{-1} and a'=xb'x^{-1}) if (and only if) for every word w in two noncommuting variables the elements w(a,a') and w(b,b') are conjugate. Using property (*) formulated above, Problem 30 asks to determine which groups G have the property that (H,G) satisfies (*) for every subgroup H of G that is generated by two elements. The above Problem 2 is a more general (and possibly more natural) version of this problem.

An automorphism \alpha\colon G\to G is class-preserving if \alpha(a) is conjuagte to a for every a\in G. If (G,G) has (*), then every class-preserving automorphism of G 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 C_1 nor to C_2. Note also that C_2 contains all abelian groups and that C_1 contains all cyclic groups.

  1. [1]
    R.D. Mauldin, The Scottish Book, Springer International Publishing, 2015. https://doi.org/10.1007/978-3-319-22897-6.
  2. [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. [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.

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