Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group G, the holomorph of G denoted Hol(G) can be described as a semidirect product or as a permutation group.

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Hol(G) as a semi-direct product

If Aut(G) is the automorphism group of G then

Hol(G)=G\rtimes Aut(G)

where the multiplication is given by

(g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta). [Eq. 1]

Typically, a semidirect product is given in the form G\rtimes_{\phi}A where G and A are groups and \phi:A\rightarrow Aut(G) is a homomorphism and where the multiplication of elements in the semi-direct product is given as

(g,a)(h,b)=(g\phi(a)(h),ab)

which is well defined, since \phi(a)\in Aut(G) and therefore \phi(a)(h)\in G.

For the holomorph, A=Aut(G) and \phi is the identity map, as such we suppress writing \phi explicitly in the multiplication given in [Eq. 1] above.

For example,

Observe, for example

(x,\sigma)(x^2,\sigma)=(x^{1%2B2\cdot2},\sigma^2)=(x^2,1)

and note also that this group is not abelian, as (x^2,\sigma)(x,\sigma)=(x,1), so that Hol(C_3) is a non-abelian group of order 6 which, by basic group theory, must be isomorphic to the symmetric group S_3.

Hol(G) as a permutation group

A group G acts naturally on itself by left and right multiplication, each giving rise to a homomorphism from G into the symmetric group on the underlying set of G. One homomorphism is defined as λ: G → Sym(G), λ(g)(h) = g·h. That is, g is mapped to the permutation obtained by left multiplying each element of G by g. Similarly, a second homomorphism ρ: G → Sym(G) is defined by ρ(g)(h) = h·g−1, where the inverse ensures that ρ(g·h)(k) = ρ(g)(ρ(h)(k)). These homomorphisms are called the left and right regular representations of G. Each homomorphism is injective, a fact referred to as Cayley's theorem.

For example, if G = C3 = {1, x, x2 } is a cyclic group of order three, then

so λ(x) takes (1, x, x2) to (x, x2, 1).

The image of λ is a subgroup of Sym(G) isomorphic to G, and its normalizer in Sym(G) is defined to be the holomorph H of G. For each f in H and g in G, there is an h in G such that f·λ(g) = λ(h)·f. If an element f of the holomorph fixes the identity of G, then for 1 in G, (f·λ(g))(1) = (λ(h)·f)(1), but the left hand side is f(g), and the right side is h. In other words, if f in H fixes the identity of G, then for every g in G, f·λ(g) = λ(f(g))·f. If g, k are elements of G, and f is an element of H fixing the identity of G, then applying this equality twice to f·λ(g)·λ(h) and once to the (equivalent) expression f·λ(g·h) gives that f(g)·f(h) = f(g·h). That is, every element of H that fixes the identity of G is in fact an automorphism of G. Such an f normalizes any λ(g), and the only λ(g) that fixes the identity is λ(1). Setting A to be the stabilizer (group theory) of the identity, the subgroup generated by A and λ(G) is semidirect product with normal subgroup λ(G) and complement A. Since λ(G) is transitive, the subgroup generated by λ(G) and the point stabilizer A is all of H, which shows the holomorph as a permutation group is isomorphic to the holomorph as semidirect product.

It is useful, but not directly relevant, that the centralizer of λ(G) in Sym(G) is ρ(G), their intersection is ρ(Z(G)) = λ(Z(G)), where Z(G) is the center of G, and that A is a common complement to both of these normal subgroups of H.

Notes

References