Zappa-Szép product

In mathematics, especially group theory, the Zappa-Szep product (also known as the knit product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products.

Internal Zappa-Szep products

Let "G" be a group with identity element "e", and let "H" and "K" be subgroups of "G". The following statements are equivalent:
* "G" = "HK" and "H" ∩ "K" = {"e"}
* For each "g" in "G", there exists a unique "h" in "H" and a unique "k" in "K" such that "g = hk". If either (and hence both) of these statements hold, then "G" is said to be an internal Zappa-Szep product of "H" and "K".

Examples

Let "G" = "GL"("n",C), the general linear group of invertible "n × n" matrices over the complex numbers. For each matrix "A" in "G", the QR decomposition asserts that there exists a unique unitary matrix "Q" and a unique upper triangular matrix "R" with positive real entries on the main diagonal such that "A" = "QR". Thus "G" is a Zappa-Szep product of the unitary group "U"("n") and the group (say) "K" of upper triangular matrices with positive diagonal entries.

One of the most important examples of this is Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa-Szep product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa-Szep product of a certain set of representatives of its Sylow subgroups.

In 1935, Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa-Szep product of the regular subgroup and a point stabilizer. He gives PSL(2,11) and the alternating group of degree 5 as examples, and of course every alternating group of prime degree is an example. This same paper gives a number of examples of groups which cannot be realized as Zappa-Szep products of proper subgroups, such as the quaternion group and the alternating group of degree 6.

External Zappa-Szep products

As with the direct and semidirect products, there is an external version of the Zappa-Szep product for groups which are not known "a priori" to be subgroups of a given group. To motivate this, let "G" = "HK" be an internal Zappa-Szep product of subgroups "H" and "K" of the group "G". For each "k" in "K" and each "h" in "H", there exist α("k","h") in "H" and β("k","h") in "K" such that "kh" = α("k","h") β("k","h"). This defines mappings α : "K" × "H" → "H" and β : "K" × "H" → "K" which turn out to have the following properties:

* For each "k" in "K", the mapping "h" $mapsto$ α("k","h") is a bijection of "H".
* For each "h" in "H", the mapping "k" $mapsto$ β("k","h") is a bijection of "K".
* α("e","h") = "h" and β("k","e") = "k" for all "h" in "H" and "k" in "K".
* α("k"₁ "k"₂, h) = α("k"₁, α("k"₂, h))
* β("k", "h"₁ "h"₂) = β(β("k", "h"₁), "h"₂)
* α("k", "h"₁ "h"₂) = α("k", "h"₁) α(β("k","h"₁),"h"₂)
* β("k"₁ "k"₂, h) = β("k"₁,α("k"₂,h)) β("k"₂,h)

for all "h"₁, "h"₂ in "H", "k"₁, "k"₂ in "K".

Turning this around, suppose "H" and "K" are groups (and let "e" denote each group's identity element) and suppose there exist mappings α : "K" × "H" → "H" and β : "K" × "H" → "K" satisfying the properties above. On the cartesian product "H" × "K", define a multiplication and an inversion mapping by, respectively,

* (h₁, k₁) (h₂, k₂) = (h₁ α(k₁,h₂), β(k₁,h₂) k₂)
* (h,k)^{− 1} = (α(k^{− 1},h^{− 1}), β(k^{− 1},h^{− 1}))

Then "H" × "K" is a group called the external Zappa-Szep product of the groups "H" and "K". The subsets "H" × {"e"} and {"e"} × "K" are subgroups isomorphic to "H" and "K", respectively, and "H" × "K" is, in fact, an internal Zappa-Szep product of "H" × {"e"} and {"e"} × "K".

Relation to semidirect and direct products

Let "G" = "HK" be an internal Zappa-Szep product of subgroups "H" and "K". If "H" is normal in "G", then the mappings α and β are given by, respectively, α("k","h") = "k h k"^{− 1} and β("k", "h") = "k". In this case, "G" is an internal semidirect product of "H" and "K".

If, in addition, "K" is normal in "G", then α("k","h") = "h". In this case, "G" is an internal direct product of "H" and "K".

References

* | year=1967, Kap. VI, §4.
* P. W. Michor, Knit products of graded Lie algebras and groups, Proceedings of the Winter School on Geometry and Physics, Srni, 1988, "Suppl. Rendiconti Circolo Matematico di Palermo", Ser. II, 22 (1989), 171-175. [http://www.arXiv.org/abs/math/9204220 ArXiv]
* G. A. Miller, Groups which are the products of two permutable proper subgroups, "Proceedings of the National Academy of Sciences" 21 (1935), 469-472. [http://www.pnas.org/content/vol21/issue7/]
* J. Szép, On the structure of groups which can be represented as the product of two subgroups, "Acta Sci. Math. Szeged" 12 (1950), 57-61.
* M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, "Comm. Algebra" 9 (1981), 841-882.
* G. Zappa, Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro, "Atti Secondo Congresso Un. Mat. Ital.", Bologna, 1940, Edizioni Cremonense, Rome, 1942, 119–125.
* A.L. Agore, A. Chirvasitu, B. Ion, G. Militaru, Factorization problems for finite groups (2007) [http://arxiv.org/abs/math/0703471 arXiv:math/0703471v2]