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

Hol(G) as a semidirect product

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

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

where the multiplication is given by

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

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

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

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

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

For example,

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

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

Observe, for example

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

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

Hol(G) as a permutation group

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

• ${\displaystyle \lambda _{x}}$ (1) = x·1 = x,
• ${\displaystyle \lambda _{x}}$ (x) = x·x = x², and
• ${\displaystyle \lambda _{x}}$ (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·${\displaystyle \lambda _{g}}$  = ${\displaystyle \lambda _{h}}$ ·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·${\displaystyle \lambda _{g}}$ )(1) = (${\displaystyle \lambda _{h}}$ ·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·${\displaystyle \lambda _{g}}$  = 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·${\displaystyle \lambda _{g}}$ ·${\displaystyle \lambda _{h}}$  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 ${\displaystyle \lambda _{G}}$ , and the only ${\displaystyle \lambda _{g}}$  that fixes the identity is λ(1). Setting A to be the stabilizer of the identity, the subgroup generated by A and ${\displaystyle \lambda _{G}}$  is semidirect product with normal subgroup ${\displaystyle \lambda _{G}}$  and complement A. Since ${\displaystyle \lambda _{G}}$  is transitive, the subgroup generated by ${\displaystyle \lambda _{G}}$  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 ${\displaystyle \lambda _{G}}$  in Sym(G) is ${\displaystyle \rho _{G}}$ , 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.

Properties

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

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