#### 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 *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(*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 *X*_{0} in terms of which the
operation *w* can be expressed, and also
at least two elements not in *X*_{0} (since
*X*_{0} is finite),
it can be deduced that the operations induced by *w* on arbitrary
objects of (*G*↓**V**) 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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