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 *g*_{1}, ..., *g _{n}*, 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(

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.

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.

And B. Sambale and G. Navarro have pointed out to me that Paul Schupp [30], 25 years before the present paper, proved an affirmative answer to the question that raised following display (2), and thus a stronger result than my Theorem 1!

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

[30]
Paul E. Schupp,
A Characterization of Inner Automorphisms,
Proc. AMS **101** (1987) p.226-228.
MR0902532.