Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that 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 formula_1, the holomorph of formula_1 denoted formula_3 can be described as a semidirect product or as a permutation group.
Hol("G") as a semidirect product.
If formula_4 is the automorphism group of formula_1 then
where the multiplication is given by
Typically, a semidirect product is given in the form formula_8 where formula_1 and formula_10 are groups and formula_11 is a homomorphism and where the multiplication of elements in the semidirect product is given as
which is well defined, since formula_13 and therefore formula_14.
For the holomorph, formula_15 and formula_16 is the identity map, as such we suppress writing formula_16 explicitly in the multiplication given in [Eq. 1] above.
For example, 
Observe, for example
and this group is not abelian, as formula_26, so that formula_27 is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group formula_28.
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" = "C"3 = {1, "x", "x"2 } is a cyclic group of order three, then
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 "N" of "G". 
For each "n" in "N" and "g" in "G", there is an "h" in "G" such that "n"·"λ"("g") = "λ"("h")·"n". If an element "n" of the holomorph fixes the identity of "G", then for 1 in "G", ("n"·"λ"("g"))(1) = ("λ"("h")·"n")(1), but the left hand side is "n"("g"), and the right side is "h". In other words, if "n" in "N" fixes the identity of "G", then for every "g" in "G", "n"·"λ"("g") = "λ"("n"("g"))·"n". If "g", "h" are elements of "G", and "n" is an element of "N" fixing the identity of "G", then applying this equality twice to "n"·"λ"("g")·"λ"("h") and once to the (equivalent) expression "n"·"λ"("g"·"h") gives that "n"("g")·"n"("h") = "n"("g"·"h"). That is, every element of "N" that fixes the identity of "G" is in fact an automorphism of "G". Such an "n" normalizes "λ"("G"), and the only "λ"("g") that fixes the identity is "λ"(1). Setting "A" to be the stabilizer 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 "N", 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 "N".