Updates to "An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange"

by George M. Bergman

Publicacions Matemàtiques 56 (2012) 91-126. 

In Proposition 1.2 of [28] (see below), V. Tolstykh shows that if  V  is any variety of groups,  G  a free group in  V  on an infinite set  X,  and  σ  an automorphism of  G  whose conjugacy class in  Aut(G)  has cardinality  ≤ card(X),  then  σ  can be expressed by a word  w  in the group operations and some constants from  G.  (Note that, conversely, any  σ  which can be so expressed has conjugacy class of cardinality  ≤ card(X);  for  w  can involve only finitely many constants from  G,  say  g1,  ..., gn,  and the conjugate of  σ  by an automorphism  α  will depend only on where  α  sends those constants, for which there are at most  card(G)n = card(X)n = card(X)  possibilities.)  Now from the fact that  w  induces an automorphism of  G,  and  G  has a set  X  of free generators, which contains a subset  X0  in terms of which the operation  w  can be expressed, and also at least two elements not in  X0  (since  X0  is finite), it can be deduced that the operations induced by  w  on arbitrary objects of  (GV)  are again automorphisms.  Thus, in the language of my paper, Tolstykh's result says that all automorphisms of  G  having small conjugacy classes are "inner with respect to  V". 

I suspect that the corresponding result can be proved similarly for many sorts of varieties  V  of algebras other than groups.  But it is not true for arbitrary varieties:   in the variety of sets (with no primitive operations), every automorphism (i.e., permutation) of a set which moves only finitely many elements has Tolstykh's small conjugacy class property; but nonidentity automorphisms of sets cannot be described by derived operations. 

In a different direction, recall that my Theorem 12 shows that for  V  the variety of Lie algebras over a field  K,  no objects of  V  have nontrivial inner automorphisms with respect to  V.  However, Nazih Nahlus has reminded me that derivations of finite dimensional Lie algebras over the real or complex numbers can be exponentiated to give automorphisms.  Since this construction respects homomorphisms, we see the inner derivations of a finite-dimensional real or complex Lie algebra  L  induce inner (in our sense) automorphisms of  L  with respect to the subcategory of finite-dimensional Lie algebras. 

Finally, Benjamin Steinberg has pointed out to me that the results of the present paper yield descriptions of the isotropy toposes, as defined in [29], of the categories of groups, k-algebras, and other structures we consider. 

Additional reference

[28] Vladimir Tolstykh, Small conjugacy classes in the automorphism groups of relatively free groups, J. Pure Appl. Algebra, 215 (2011) 2086-2098.  MR 2012f:20108. 

[29] Jonathon Funk, Pieter Hofstra and Benjamin Steinberg, Isotropy and crossed toposes, Theory Appl. Categ. 26 (2012) 660-709.  MR3065939. 

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