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

Hol( G ) as a semidirect product

If Aut ⁡ ( G ) {\displaystyle \operatorname {Aut} (G)} is the automorphism group of G {\displaystyle G}, then

Hol ⁡ ( G ) = G ⋊ Aut ⁡ ( G ) {\displaystyle \operatorname {Hol} (G)=G\rtimes \operatorname {Aut} (G)},

where the multiplication is given by

( g , α ) ( h , β ) = ( g α ( h ) , α β ) . {\displaystyle (g,\alpha )(h,\beta )=(g\alpha (h),\alpha \beta ).}

Typically, a semidirect product is given in the form G ⋊ ϕ A {\displaystyle G\rtimes _{\phi }A}, where G {\displaystyle G} and A {\displaystyle A} are groups and ϕ : A → Aut ⁡ ( G ) {\displaystyle \phi :A\rightarrow \operatorname {Aut} (G)} is a homomorphism, and where the multiplication of elements in the semidirect product is given as

( g , a ) ( h , b ) = ( g ϕ ( a ) ( h ) , a b ) {\displaystyle (g,a)(h,b)=(g\phi (a)(h),ab)}.

This is well defined since ϕ ( a ) ∈ Aut ⁡ ( G ) {\displaystyle \phi (a)\in \operatorname {Aut} (G)}, and therefore ϕ ( a ) ( h ) ∈ G {\displaystyle \phi (a)(h)\in G}.

For the holomorph, A = Aut ⁡ ( G ) {\displaystyle A=\operatorname {Aut} (G)} and ϕ {\displaystyle \phi } is the identity map. As such, we suppress writing ϕ {\displaystyle \phi } explicitly in the multiplication given in equation (1) above.

As an example, take

  • G = C 3 = ⟨ x ⟩ = { 1 , x , x 2 } {\displaystyle G=C_{3}=\langle x\rangle =\{1,x,x^{2}\}} the cyclic group of order 3,
  • Aut ⁡ ( G ) = ⟨ σ ⟩ = { 1 , σ } {\displaystyle \operatorname {Aut} (G)=\langle \sigma \rangle =\{1,\sigma \}}, where σ ( x ) = x 2 {\displaystyle \sigma (x)=x^{2}}, and
  • Hol ⁡ ( G ) = { ( x i , σ j ) } {\displaystyle \operatorname {Hol} (G)=\{(x^{i},\sigma ^{j})\}} with the multiplication given by:

( x i 1 , σ j 1 ) ( x i 2 , σ j 2 ) = ( x i 1 + i 2 2 j 1 , σ j 1 + j 2 ) {\displaystyle (x^{i_{1}},\sigma ^{j_{1}})(x^{i_{2}},\sigma ^{j_{2}})=(x^{i_{1}+i_{2}2^{^{j_{1}}}},\sigma ^{j_{1}+j_{2}})}, where the exponents of x {\displaystyle x} are taken mod 3 and those of σ {\displaystyle \sigma } mod 2.

Observe that

( x , σ ) ( x 2 , σ ) = ( x 1 + 2 ⋅ 2 , σ 2 ) = ( x 2 , 1 ) {\displaystyle (x,\sigma )(x^{2},\sigma )=(x^{1+2\cdot 2},\sigma ^{2})=(x^{2},1)} while ( x 2 , σ ) ( x , σ ) = ( x 2 + 1 ⋅ 2 , σ 2 ) = ( x , 1 ) {\displaystyle (x^{2},\sigma )(x,\sigma )=(x^{2+1\cdot 2},\sigma ^{2})=(x,1)}.

Hence, this group is not abelian, and so Hol ⁡ ( C 3 ) {\displaystyle \operatorname {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 {\displaystyle 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), λg(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 ρg(h) = h·g−1, where the inverse ensures that ρgh(k) = ρg(ρh(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

  • λx(1) = x·1 = x,
  • λx(x) = x·x = x2, and
  • λx(x2) = x·x2 = 1,

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 N of G. For each n in N and g in G, there is an h in G such that n·λg = λh·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·λg)(1) = (λh·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·λg = λn(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·λg·λh and once to the (equivalent) expression n·λgg 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 λG, and the only λg that fixes the identity is λ(1). Setting A to be the stabilizer of the identity, the subgroup generated by A and λG is semidirect product with normal subgroup λG and complement A. Since λG is transitive, the subgroup generated by λG 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 λG in Sym(G) is ρG, their intersection is ρ Z ( G ) = λ Z ( G ) {\displaystyle \rho _{Z(G)}=\lambda _{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.

Properties

  • ρ(G) ∩ Aut(G) = 1
  • Aut(G) normalizes ρ(G) so that canonically ρ(G)Aut(G) ≅ G ⋊ Aut(G)
  • Inn ⁡ ( G ) ≅ Im ⁡ ( g ↦ λ ( g ) ρ ( g ) ) {\displaystyle \operatorname {Inn} (G)\cong \operatorname {Im} (g\mapsto \lambda (g)\rho (g))} since λ(g)ρ(g)(h) = ghg−1 (Inn ⁡ ( G ) {\displaystyle \operatorname {Inn} (G)} is the group of inner automorphisms of G.)
  • K ≤ G is a characteristic subgroup if and only if λ(K) ⊴ Hol(G)