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 automorphisms in a uniform context. In group theory, for a group $G$ , the holomorph of $G$ denoted $\operatorname {Hol} (G)$ can be described as a semidirect product or as a permutation group.

Hol(G) as a semi-direct 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 ).

[Eq. 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 semi-direct 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 [Eq. 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 note also that 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), 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 = C₃ = {1, x, x² } is a cyclic group of order three, then

λ(x)(1) = x·1 = x,
λ(x)(x) = x·x = x², and
λ(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 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, h 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). That is, every element of H that fixes the identity of G is in fact an automorphism of G. Such an f normalizes 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.

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
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

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