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 [math]\displaystyle{ G }[/math], the holomorph of [math]\displaystyle{ G }[/math] denoted [math]\displaystyle{ \operatorname{Hol}(G) }[/math] can be described as a semidirect product or as a permutation group.

Hol(G) as a semidirect product

If [math]\displaystyle{ \operatorname{Aut}(G) }[/math] is the automorphism group of [math]\displaystyle{ G }[/math] then

[math]\displaystyle{ \operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G) }[/math]

where the multiplication is given by

[math]\displaystyle{ (g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta). }[/math] [Eq. 1]

Typically, a semidirect product is given in the form [math]\displaystyle{ G\rtimes_{\phi}A }[/math] where [math]\displaystyle{ G }[/math] and [math]\displaystyle{ A }[/math] are groups and [math]\displaystyle{ \phi:A\rightarrow \operatorname{Aut}(G) }[/math] is a homomorphism and where the multiplication of elements in the semidirect product is given as

[math]\displaystyle{ (g,a)(h,b)=(g\phi(a)(h),ab) }[/math]

which is well defined, since [math]\displaystyle{ \phi(a)\in \operatorname{Aut}(G) }[/math] and therefore [math]\displaystyle{ \phi(a)(h)\in G }[/math].

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

For example,

• [math]\displaystyle{ G=C_3=\langle x\rangle=\{1,x,x^2\} }[/math] the cyclic group of order 3
• [math]\displaystyle{ \operatorname{Aut}(G)=\langle \sigma\rangle=\{1,\sigma\} }[/math] where [math]\displaystyle{ \sigma(x)=x^2 }[/math]
• [math]\displaystyle{ \operatorname{Hol}(G)=\{(x^i,\sigma^j)\} }[/math] with the multiplication given by:

[math]\displaystyle{ (x^{i_1},\sigma^{j_1})(x^{i_2},\sigma^{j_2}) = (x^{i_1+i_22^{^{j_1}}},\sigma^{j_1+j_2}) }[/math] where the exponents of [math]\displaystyle{ x }[/math] are taken mod 3 and those of [math]\displaystyle{ \sigma }[/math] mod 2.

Observe, for example

[math]\displaystyle{ (x,\sigma)(x^2,\sigma)=(x^{1+2\cdot2},\sigma^2)=(x^2,1) }[/math]

and this group is not abelian, as [math]\displaystyle{ (x^2,\sigma)(x,\sigma)=(x,1) }[/math], so that [math]\displaystyle{ \operatorname{Hol}(C_3) }[/math] is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group [math]\displaystyle{ S_3 }[/math].

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), [math]\displaystyle{ \lambda_g }[/math](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 [math]\displaystyle{ \rho_g }[/math](h) = h·g⁻¹, where the inverse ensures that [math]\displaystyle{ \rho_{gh} }[/math](k) = [math]\displaystyle{ \rho_g }[/math]([math]\displaystyle{ \rho_h }[/math](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

• [math]\displaystyle{ \lambda_x }[/math](1) = x·1 = x,
• [math]\displaystyle{ \lambda_x }[/math](x) = x·x = x², and
• [math]\displaystyle{ \lambda_x }[/math](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·[math]\displaystyle{ \lambda_g }[/math] = [math]\displaystyle{ \lambda_h }[/math]·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·[math]\displaystyle{ \lambda_g }[/math])(1) = ([math]\displaystyle{ \lambda_h }[/math]·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·[math]\displaystyle{ \lambda_g }[/math] = [math]\displaystyle{ \lambda_{n(g)} }[/math]·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·[math]\displaystyle{ \lambda_g }[/math]·[math]\displaystyle{ \lambda_h }[/math] and once to the (equivalent) expression n·[math]\displaystyle{ \lambda_{gh} }[/math] 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 [math]\displaystyle{ \lambda_G }[/math], and the only [math]\displaystyle{ \lambda_g }[/math] that fixes the identity is λ(1). Setting A to be the stabilizer of the identity, the subgroup generated by A and [math]\displaystyle{ \lambda_G }[/math] is semidirect product with normal subgroup [math]\displaystyle{ \lambda_G }[/math] and complement A. Since [math]\displaystyle{ \lambda_G }[/math] is transitive, the subgroup generated by [math]\displaystyle{ \lambda_G }[/math] 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 [math]\displaystyle{ \lambda_G }[/math] in Sym(G) is [math]\displaystyle{ \rho_G }[/math], their intersection is [math]\displaystyle{ \rho_{Z(G)}=\lambda_{Z(G)} }[/math], 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)
• [math]\displaystyle{ \operatorname{Inn}(G)\cong \operatorname{Im}(g\mapsto \lambda(g)\rho(g)) }[/math] since λ(g)ρ(g)(h) = ghg⁻¹ ([math]\displaystyle{ \operatorname{Inn}(G) }[/math] is the group of inner automorphisms of G.)
• K ≤ G is a characteristic subgroup if and only if λ(K) ⊴ Hol(G)

References

• Hall, Marshall, Jr. (1959), The theory of groups, Macmillan 
• Burnside, William (2004), Theory of Groups of Finite Order, 2nd ed., Dover, p. 87