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 ${\displaystyle G}$, 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 semi-direct product

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

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 semi-direct product is given as

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

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,

${\displaystyle (x^{i_{1}},\sigma ^{j_{1}})(x^{i_{2}},\sigma ^{j_{2}})=(x^{i_{1}+i_{2}2^{^{j_{1}}}},\sigma ^{j_{1}+j_{2}})}$ 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 note also that 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

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(hf. If an element f of the holomorph fixes the identity of G, then for 1 in G, (f·λ(g))(1) = (λ(hf)(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(gλ(h) and once to the (equivalent) expression f·λ(g·h) gives that f(gf(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.

References

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