gr.group theory - Uses of the holomorph, Hol($G$) = $G rtimes $ Aut($G$)

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.