Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group $G$, denoted $\operatorname {Hol} (G)$, is a group that simultaneously contains (copies of) $G$ and its automorphism group $\operatorname {Aut} (G)$. 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 $\operatorname {Aut} (G)$ is the automorphism group of $G$ then

$\operatorname {Hol} (G)=G\rtimes \operatorname {Aut} (G)$

where the multiplication is given by

$(g,\alpha )(h,\beta )=(g\alpha (h),\alpha \beta ).$ (1)

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

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

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

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

For example,

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

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

Observe, for example

$(x,\sigma )(x^{2},\sigma )=(x^{1+2\cdot 2},\sigma ^{2})=(x^{2},1)$

and this group is not abelian, as $(x^{2},\sigma )(x,\sigma )=(x,1)$, so that $\operatorname {Hol} (C_{3})$ is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group $S_{3}$.

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), $\lambda _{g}$(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 $\rho _{g}$(h) = h·g⁻¹, where the inverse ensures that $\rho _{gh}$(k) = $\rho _{g}$($\rho _{h}$(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

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

References

• Hall, Marshall Jr. (1959), The theory of groups, Macmillan, MR 0103215
• Burnside, William (2004), Theory of Groups of Finite Order, 2nd ed., Dover, p. 87