Is there a name for this group?

[David Harden]

convention: action is on the right, so gh means 'do g, and then do h';
likewise, the result of conjugating x through g is x^g = g^(-1)xg

Let G be a finite group. G is isomorphic to a group of permutations of
its own elements, via the Cayley isomorphism g ===> ( x ---> xg ). The
normalizer, in S_|G|, of this subgroup isomorphic to G, is the
holomorph Hol(G), which is obtained by adjoining the permutations of G
coming from automorphisms of G. The resulting group consists of all
permutations f of G of the form x^f = (x^alpha)g, where alpha is in
Aut(G) and g is in G.

But this is not the end of the story. In what follows, G is assumed to
be nonabelian.
Since G is nonabelian, antiautomorphisms of G are not automorphisms of
G. Furthermore, it is always true that G possesses antiautomorphisms,
since the inversion map ( x ---> x^(-1) ) from G to itself is one of
them. Since the composition of two antiautomorphisms of G results in
an automorphism of G, adjoining one of these to Hol(G) results in the
adjunction of all of them. Thus adjoining inversion results in
adjunction of all of them.
Right-multiplying an element of Hol(G) by inversion results in a
permutation of G of the form
(x)f = g^(-1)(x^alpha)^(-1) = g^(-1)*(x^(-1))^alpha =
g^(-1)*((x^(-1))^alpha)*g*g^(-1) = (((x^(-1))^alpha)^g)*g^(-1), which
is the result of left-multiplying an element of Hol(G) by inversion.
Let N be the group obtained by adjoining inversion to Hol(G). The
above calculation implies directly that Hol(G) is normal in N, and
that [N: Hol(G)] = 2 (which also implies Hol(G) is normal in N). Does
N have a name? If so, what is it, who coined it, and in what reference
did this author coin it?

---- David

[ma...@mimosa.csv.warwick.ac.uk]

In article <83fa8127-4935-4b3c...@c33g2000hsd.googlegroups.com>,

My thought was that the answer to your question is "sci.math"!

I have come across the group you define, but I have never heard of a name
for it. Hol(G) = G.Aut(G) is the normalizer of G (identified with the right
regular representation of G) in the symmetric group on G. The centralizer C
of G is the image of the left regular representation of G, and also acts
regularly on the elements of G. Hol(G) also normalizes C. Your extra
involution acting by inversion on the group elements interchanges the two
regular subgroups G and C.

There was not need for you to ssume that G is finite - this is all valid
for an arbitrary nonabelian group G.

Derek Holt.

[David Harden]

On Mar 9, 1:36 pm, ma...@mimosa.csv.warwick.ac.uk () wrote:
> In article <83fa8127-4935-4b3c-92fc-0c1952ad8...@c33g2000hsd.googlegroups.com>,

Then, in light of the fact that the index of the holomorph in this
group
is 2, and to make light of the similarity of the word "holomorph"
with
a term in analysis, I suggest that it be called the biholomorph.
Are there any takers for this term?

>
> I have come across the group you define, but I have never heard of a name
> for it. Hol(G) = G.Aut(G) is the normalizer of G (identified with the right
> regular representation of G) in the symmetric group on G. The centralizer C
> of G is the image of the left regular representation of G, and also acts
> regularly on the elements of G. Hol(G) also normalizes C. Your extra
> involution acting by inversion on the group elements interchanges the two
> regular subgroups G and C.
>

> There was not need for you to assume that G is finite - this is all valid

> for an arbitrary nonabelian group G.

Thanks for the reminder. I needed it. But a tantalizing context in
which I
have considered applying this is most definitely finite, and that,
along
with other sources of my overall finite bias in group theory, has
manifested itself in the way I discussed this.

A few other words:

I posted this to the Wikipedia discussion page for "holomorph" and a
respondent there said his computations showed that the biholomorph of
G wasn't always the full normalizer of Hol(G) in S_|G|. Some examples
of groups G where the index of Hol(G) in its S_|G|-normalizer is > 2
are, the dihedral, semidihedral, and quaternionic groups of order 16,
along with C_4 x S_3 and the dihedral group of order 24.

There, the notation Ant(G) is used for the thing I am calling the
biholomorph.

---- David

[ma...@mimosa.csv.warwick.ac.uk]

In article <e0ca28d0-2573-435c...@p73g2000hsd.googlegroups.com>,

That's right! I was also wondering whether the biholomorph was the full
normalizer in S_|G| of GC, but that is not true either in the above
examples. I believe that when G is a perfect group (i.e. G = [G,G]),
G and C are the only regular normal subgroups of GC, so in that case
the biholomorph will be the normalizer in S_|G| of G. I have not thought
about whether it is also necessarily the fullnormalizer of Hol(G) in
that case.

Derek Holt.