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 automorphism in a uniform context.

Hol("G") as a semi-direct product

In group theory, for a group $G$, the holomorph of $G$ denoted $Hol(G)$ can be described in two ways. First, if $Aut(G)$ is the automorphism group of $G$ then:$Hol(G)=G\; times\; Aut(G)$where the multiplication is given by: [Eq. 1]

Typically, a semi-direct product is given in the form $G\; times\_\{phi\}A$ where $G$ and $A$ are groups and $phi:A\; ightarrow\; Aut(G)$ is a homomorphism and where the multiplication of elements in the semi-direct product is given as:$(g,a)(h,b)=(gphi(a)(h),ab)$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,
* $G=C\_3=langle\; x\; angle=\{1,x,x^2\}$ the cyclic group of order 3
* $Aut(G)=langle\; sigma\; angle=\{1,sigma\}$ where $sigma(x)=x^2$
* $Hol(G)=\{(x^i,sigma^j)\}$ with the multiplication given by
* $(x^\{i\_1\},sigma^\{j\_1\})(x^\{i\_2\},sigma^\{j\_2\})=(x^\{i\_1+i\_22^\{^\{j\_1\},sigma^\{j\_1+j\_2\})$where the exponents of $x$ are taken mod 3 and those of $sigma$ mod 2.

Observe, for example:$(x,sigma)(x^2,sigma)=(x^\{1+2cdot2\},sigma^2)=(x^2,1)$and note also that this group is not abelian, as , 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 are 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" = "C"₃ = {1, "x", "x"² } is a cyclic group of order three, then
* "λ"("x")(1) = "x"·1 = "x",
* "λ"("x")("x") = "x"·"x" = "x"², and
* "λ"("x")("x"²) = "x"·"x"² = 1,so "λ"("x") takes (1, "x", "x"²) to ("x", "x"², 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"). In other words, 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

* "ρ"("G") ∩ Aut("G") = 1
* Aut("G") normalizes "ρ"("G") so that canonically "ρ"("G")Aut("G") ≅ "G" ⋊ Aut("G")
*$Inn(G)cong\; Im(gmapsto\; lambda(g)\; ho(g))$ since "λ"("g")"ρ"("g")("h") = "ghg"^-1
* "K" ≤ "G" is a characteristic subgroup if and only if λ(K) ⊴ Hol("G")

References

* | year=1959