> < ^ Date: Mon, 13 Dec 1999 15:46:35 +0100 (MEZ)
> < ^ From: Mathias Kratzer <kratzer@exp-math.uni-essen.de >
> ^ Subject: ConjugatorAutomorphism

Dear GAP-Forum,

in Chapter 35, Section 5 the GAP4-manual says:

> * ConjugatorAutomorphism( G, g ) O
>
> creates for g in the same Family as the elements of G
> the automorphism of G defined by h |--> h^{elm} for all h\in G .
~~~~~~~~~~~~~~~~~~~~~
Reading these lines I guess that 'ConjugatorAutomorphism' was imple-
mented to sort of generalize the function 'InnerAutomorphism': The
latter one only provides access to automorphisms of a group G which
are induced by elements of G; by means of the first one you can also
construct "conjugations" by elements of e. g. a supergroup of G.

Of course, the mapping h |--> h^g, for g \in S, S a supergroup of G,
is an automorphism of G if and only if G is invariant under conjuga-
tion by g.
Surprisingly, the manual does not keep its readers aware of this fact
at all, and --- confer the example below --- 'ConjugatorAutomorphism'
does not even give you a warning let alone an error message, when you
try to induce an automorphism from an element which violates the
invariance condition stated above:

gap> G := Group( [ (1,2,3,4) ] );
Group([ (1,2,3,4) ])
gap> phi := ConjugatorAutomorphism( G, (1,2) );
^(1,2)
gap> eltsG := Elements( G );
[ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) ]
gap> List(eltsG, elt -> elt^phi);
[ (), (1,3,4,2), (1,4)(2,3), (1,2,4,3) ]
gap> IsConjugatorAutomorphism(phi);
true
gap>


Would the function 'ConjugatorAutomorphism' not better be called 'Con-
jugatorISOmorphism'? Or is there anything I've just misunderstood?

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