Cassie's law

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 $\mathrm{Hol}(G)$ can be described as a semidirect product or as a permutation group.

Hol(G) as a semi-direct product

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

$\mathrm{Hol}(G)=G⋊\mathrm{Aut}(G)$

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 $G{⋊}_{\varphi }A$ where $G$ and $A$ are groups and $\varphi :A\to \mathrm{Aut}(G)$ is a homomorphism and where the multiplication of elements in the semi-direct product is given as

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

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

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

For example,

• $G=C_3=⟨x⟩=\{1,x,x^2\}$ the cyclic group of order 3
• $\mathrm{Aut}(G)=⟨\sigma ⟩=\{1,\sigma \}$ where $\sigma (x)=x^2$
• $\mathrm{Hol}(G)=\{(x^i,{\sigma }^j)\}$ 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 $\mathrm{Hol}(C_3)$ 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 any 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.

Notes

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

References

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