Zappa-Szep product
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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.
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[edit] 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.
[edit] Example
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.
[edit] 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
α(k,h) is a bijection of H. - For each h in H, the mapping k 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(k1 k2, h) = α(k1, α(k2, h))
- β(k, h1 h2) = β(β(k, h2), h1)
- α(k, h1 h2) = α(k, h1) α(β(k,h1),h2)
- β(k1 k2, h) = β(k1,α(k2,h)) β(k2,h)
for all h1, h2 in H, k1, k2 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.
[edit] 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.
[edit] References
- 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. ArXiv: math.GR/9204220.
- 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.
- 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.
