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In every group theory textbook I've read, the holomorph has been defined, and maybe a few problems done with it. I've also seen papers focusing on computing Hol($G$) for a specific class of $G$.

One thing I have never seen is any actual use for it. Are there major results using the holomorph of a group? Does it occur in the proof of any useful theorems?

It seems intrisically interesting to me since it allows you to treat automorphisms of a group and elements of a group uniformly, and I would definitely like to learn more about it.

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Upvoted for appropriate and good motivation for a response. I have been wondering this as well. –  Quadrescence Feb 3 '10 at 0:55
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up vote 7 down vote accepted

If G is abelian, then the holomorph of G is a reasonably nice group. If G is a finite elementary abelian p-group of order pn, then you can consider it to be a vector space over Z/pZ. The automorphism group is the group GL(n,p) of invertible n×n matrices over Z/pZ. The holomorph is called the affine general linear group, AGL(n,p), which can be thought of as (n+1)×(n+1) matrices [ A, v ; 0, 1 ] where A in Aut(G) ≅ GL(n,p) and v in G ≅ (Z/pZ)^n. If you restrict the automorphism group to only include GF(p^k) automorphisms, where k divides n, then you get a subgroup AGL(n/k,p^k) that is also important.

These sorts of groups are (one of) the standard examples of primitive permutation groups. Every soluble primitive permutation group has some minimal normal subgroup G that is elementary abelian, and a maximal subgroup M contained in Aut(G) = GL(n,p) that acts irreducibly on G, and the group itself is then the obvious subgroup { [ A, v ; 0, 1 ] : A in M } of AGL(n,p). Insoluble primitive groups can replace G with a non-abelian simple group or two, but a fair amount of the theory still applies.

These are all examples of the original motivation of the holomorph as the normalizer in the symmetric group of the regular representation of the group. For instance, a Sylow p-subgroup of the symmetric group on p points is regular of order p, and the Sylow normalizer is the holomorph, AGL(1,p).

Regular normal subgroups occur frequently in permutation groups and computational group theory (usually as something to be avoided due to behaving so differently), and their normalizers (aka, the whole group, since the subgroup is normal), are contained in the holomorph.

Primitive soluble groups, and in general, "irreducible" subgroups of AGL(n,p), tend to be important "boundary" examples (as in the boundary of a Schunck class) which do not have a property, but such that every quotient does. "M" is chosen to have the property, and then "G" is taken to be an irreducible M-module such that M⋉G does not have the property. It bugs me to call M the group and G the module, so in the next part M will be the module, and R the ring:

Something similar to a holomorph can be constructed from any module over a ring. You take the matrices [ r, m ; 0, 1 ] where r in R, m in M, and you get another ring where M the module becomes M the ideal; a so-called trivial extension. If instead of all of R, you just take the units of R, GL(1,R), then you get a nice group. For instance, taking R to be the p-adic integers extended by a p'th root of unity z (not already in there), and M to be R, then you get a very important pro-p-group of coclass 1, G = { [ z^i, r ; 0, 1 ] : 0 ≤ i < p, r in R }. For p=2, this is a pro-2 version of the dihedral group, and for all p its finite quotients are "mainline" p-groups of maximal class.

When G is not abelian, many of these comments still apply, but the matrix formulations are usually less enlightening. In general, the holomorph is a very nice setting in which to work with a group G and its automorphisms, with a regular normal subgroup G, with a primitive soluble group, or with various other nice examples.

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The holomorph comes up naturally in one of my papers. See Proposition 2.2 in the arXiv version of this paper: A family of embedding spaces. The result is that the space of embeddings of $S^j$ in $\mathbb R^n$ has the homotopy-type of a bundle:

$$ SO_n \times_{SO_{n-j}} (C \rtimes K_{n,j})$$

Here $K_{n,j}$ is the space of embeddings of $\mathbb R^j$ in $\mathbb R^n$ which are a fixed linear inclusion outside of a fixed ball. $C \rtimes K_{n,j}$ is the fiber bundle over $K_{n,j}$ whose fiber over a point $f \in K_{n,j}$ is the complement of the image of $f$, i.e. $\mathbb R^n \setminus img(f)$.

The case $n=3, j=1$ is especially nice as knot complements are known to be $K(\pi,1)$ spaces and the space $K_{3,1}$ has components that are $K(\pi,1)$ spaces. So $C \rtimes K_{3,1}$ is the union of classifying spaces of holomorphs. It turns out those holomorphs are for the fundamental groups of knot complements in $S^3$. (technically, you're not using the full automorphism group of the fundamental group of the knot complement, but the one that fixes the peripheral structure -- the meridian and longitude of the knot).

So in a sense $C \rtimes K_{n,j}$ is a type of `space-level' holomorph. It's philosophically similar to the construction of tautological bundles over Grassmannians.

I remember Maria Nogin (Voloshina) did work on holomorph groups in her dissertation.

That's everything on the top of my head.

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The holomorph is used in the theory of Hopf Galois structures. If L/K is a Galois extension with Galois group G, then isomorphism classes of K-Hopf algebras H such that there is an L-Hopf algebra isomorphism $L \otimes H \to L[G]$ are in bijection with regular embeddings of G into its holomorph. This is a special case of a more general statement for arbitrary separable extensions and group rings for other groups of the same order due to Greither and Pareigis, with enhancements by Byott.

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Another extensive use of the holomorph is in the study of crossed modules / 2-groups (in the categorification sense). Any group yields a one object groupoid. Any groupoid $G$ yields an endomorphism gadget $G^G$, that is the category of functors from $G$ to itself. This is at one and the same time a groupoid and a monoid (under composition of functors). Now look at the functors that are isomorphisms. That gives you a category which is also a group. (Yes I do mean that! It is the subgroup of the monoid structure.) That makes it a 2-group in that categorified sense. Any 2-group yield a crossed module and the crossed module here is just the inner automorphism homomorphism $G\to Aut(G)$. The 2-group is the holomorph!

This basic construction is central to many treatments of non-Abelian cohomology (of groups, and of sheaves of groups.) See Larry Breen's papers, and more recently papers by Aldrovandi and Noohi. (You can find stuff on this in the n-Lab and also in my Menagerie notes, a version of which is on my n-Lab homepage.) It is also central to attempts to implement Grothendieck's Pursuing Stack programme although that needs an enormous amount more input as well!

(I can give detailed references if they are of interest.)

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Aha! I suspected this, but never sat down and tried to convince myself of it. Neat! –  Eric Peterson Feb 3 '10 at 15:48
    
Yeah I'd definitely appreciate some references if you wouldn't mind. –  Adam Libster Feb 4 '10 at 8:11
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The references for Breen include the excellent: L. Breen, 2006, Notes on 1- and 2-gerbes , arxiv.org/abs/math/0106083v1. but that is really just for the general ideas of non-Abelian cohomology. For Aldrovandi and Noohi here is the Google scholar Bibtex ref. (note part 2 is available as a preprint) title={{Butterflies I: Morphisms of 2-group stacks}}, author={Aldrovandi, E. and Noohi, B.}, journal={Advances in Mathematics}, volume={221}, number={3}, pages={687--773}, year={2009}, publisher={Elsevier} } I'll continue in a second comment. –  Tim Porter Feb 4 '10 at 14:43
    
I notice that in my Menagerie notes and also in the n-Lab entries on these things, the proof of that link is either omitted or left as an exercise. I will write up a simple version for my notes, adapt it for the n-Lab and (hopefully) then put a version here. The Menagerie notes are there for consultation. Visit ncatlab.org/timporter/show/HomePage and follow links. Feed back is welcome ... as long as it is excellent!!! –  Tim Porter Feb 4 '10 at 14:47
    
OOPS. The Breen link seems to be to the wrong paper. It should be arxiv.org/abs/math/0611317 –  Tim Porter Feb 4 '10 at 14:49
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