In mathematics, especially group theory, the Zappa–Szép product (also known as the Zappa–Rédei–Szép product, general product, knit product, exact factorization or bicrossed product) describes a way in which a group can be constructed from two subgroups. It is a generalization of the direct and semidirect products. It is named after Guido Zappa (1940) and Jenő Szép (1950) although it was independently studied by others including B.H. Neumann (1935), G.A. Miller (1935), and J.A. de Séguier (1904).[1]
One of the most important examples of this is Philip Hall's 1937 theorem on the existence of Sylow systems for soluble groups. This shows that every soluble group is a Zappa–Szép product of a Hall p'-subgroup and a Sylow p-subgroup, and in fact that the group is a (multiple factor) Zappa–Szép product of a certain set of representatives of its Sylow subgroups.
In 1935, George Miller showed that any non-regular transitive permutation group with a regular subgroup is a Zappa–Szép 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–Szép products of proper subgroups, such as the quaternion group and the alternating group of degree 6.
External Zappa–Szép products
As with the direct and semidirect products, there is an external version of the Zappa–Szép 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–Szép 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:
α(e, h) = h and β(k, e) = k for all h in H and k in K.
α(k1k2, h) = α(k1, α(k2, h))
β(k, h1h2) = β(β(k, h1), h2)
α(k, h1h2) = α(k, h1) α(β(k, h1), h2)
β(k1k2, h) = β(k1, α(k2, h)) β(k2, h)
for all h1, h2 in H, k1, k2 in K. From these, it follows that
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.
(Indeed, suppose α(k, h1) = α(k, h2). Then h1 = α(k−1k, h1) = α(k−1, α(k, h1)) = α(k−1, α(k, h2)) = h2. This establishes injectivity, and for surjectivity, use h = α(k, α(k−1, h)).)
More concisely, the first three properties above assert the mapping α : K × H → H is a left action of K on (the underlying set of) H and that β : K × H → K is a right action of H on (the underlying set of) K. If we denote the left action by h → kh and the right action by k → kh, then the last two properties amount to k(h1h2) = kh1kh1h2 and (k1k2)h = k1k2hk2h.
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 productH × 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–Szép product of the groups H and K. The subsetsH × {e} and {e} × K are subgroups isomorphic to H and K, respectively, and H × K is, in fact, an internal Zappa–Szép product of H × {e} and {e} × K.
Relation to semidirect and direct products
Let G = HK be an internal Zappa–Szép 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. This is easy to see because and since by normality of , . 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
^Martin W. Liebeck; Cheryl E. Praeger; Jan Saxl (2010). Regular Subgroups of Primitive Permutation Groups. American Mathematical Soc. pp. 1–2. ISBN978-0-8218-4654-4.
Michor, P. W. (1989), "Knit products of graded Lie algebras and groups", Proceedings of the Winter School on Geometry and Physics, Srni, Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II, 22: 171–175, arXiv:math/9204220, Bibcode:1992math......4220M.
Miller, G. A. (1935), "Groups which are the products of two permutable proper subgroups", Proceedings of the National Academy of Sciences, 21 (7): 469–472, Bibcode:1935PNAS...21..469M, doi:10.1073/pnas.21.7.469, PMC1076628, PMID 16588002
Szép, J. (1950), "On the structure of groups which can be represented as the product of two subgroups", Acta Sci. Math. Szeged, 12: 57–61.
Takeuchi, M. (1981), "Matched pairs of groups and bismash products of Hopf algebras", Comm. Algebra, 9 (8): 841–882, doi:10.1080/00927878108822621.
Zappa, G. (1940), "Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro", Atti Secondo Congresso Un. Mat. Ital., Bologna; Edizioni Cremonense, Rome, (1942) 119–125.
Agore, A.L.; Chirvasitu, A.; Ion, B.; Militaru, G. (2007), Factorization problems for finite groups, arXiv:math/0703471, Bibcode:2007math......3471A, doi:10.1007/s10468-009-9145-6, S2CID 18024087.
Brin, M. G. (2005). "On the Zappa-Szép Product". Communications in Algebra. 33 (2): 393–424. arXiv:math/0406044. doi:10.1081/AGB-200047404. S2CID 15169734.
January 24, 2023
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In mathematics especially group theory the Zappa Szep product also known as the Zappa Redei Szep product general product knit product exact factorization or bicrossed product describes a way in which a group can be constructed from two subgroups It is a generalization of the direct and semidirect products It is named after Guido Zappa 1940 and Jeno Szep 1950 although it was independently studied by others including B H Neumann 1935 G A Miller 1935 and J A de Seguier 1904 1 Contents 1 Internal Zappa Szep products 2 Examples 3 External Zappa Szep products 4 Relation to semidirect and direct products 5 ReferencesInternal Zappa Szep products EditLet 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 EditLet 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 Philip 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 George 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 EditAs 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 a k h in H and b k h in K such that kh a k h b k h This defines mappings a K H H and b K H K which turn out to have the following properties a e h h and b k e k for all h in H and k in K a k1k2 h a k1 a k2 h b k h1h2 b b k h1 h2 a k h1h2 a k h1 a b k h1 h2 b k1k2 h b k1 a k2 h b k2 h for all h1 h2 in H k1 k2 in K From these it follows that For each k in K the mapping h a k h is a bijection of H For each h in H the mapping k b k h is a bijection of K Indeed suppose a k h1 a k h2 Then h1 a k 1k h1 a k 1 a k h1 a k 1 a k h2 h2 This establishes injectivity and for surjectivity use h a k a k 1 h More concisely the first three properties above assert the mapping a K H H is a left action of K on the underlying set of H and that b K H K is a right action of H on the underlying set of K If we denote the left action by h kh and the right action by k kh then the last two properties amount to k h1h2 kh1 kh1h2 and k1k2 h k1k2h k2h Turning this around suppose H and K are groups and let e denote each group s identity element and suppose there exist mappings a K H H and b 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 a k1 h2 b k1 h2 k2 h k 1 a k 1 h 1 b 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 EditLet G HK be an internal Zappa Szep product of subgroups H and K If H is normal in G then the mappings a and b are given by respectively a k h k h k 1 and b k h k This is easy to see because h 1 k 1 h 2 k 2 h 1 k 1 h 2 k 1 1 k 1 k 2 displaystyle h 1 k 1 h 2 k 2 h 1 k 1 h 2 k 1 1 k 1 k 2 and h 1 k 1 h 2 k 1 1 H displaystyle h 1 k 1 h 2 k 1 1 in H since by normality of H displaystyle H k 1 h 2 k 1 1 H displaystyle k 1 h 2 k 1 1 in H In this case G is an internal semidirect product of H and K If in addition K is normal in G then a k h h In this case G is an internal direct product of H and K References Edit Martin W Liebeck Cheryl E Praeger Jan Saxl 2010 Regular Subgroups of Primitive Permutation Groups American Mathematical Soc pp 1 2 ISBN 978 0 8218 4654 4 Huppert B 1967 Endliche Gruppen in German Berlin New York Springer Verlag ISBN 978 3 540 03825 2 MR 0224703 OCLC 527050 Kap VI 4 Michor P W 1989 Knit products of graded Lie algebras and groups Proceedings of the Winter School on Geometry and Physics Srni Suppl Rendiconti Circolo Matematico di Palermo Ser II 22 171 175 arXiv math 9204220 Bibcode 1992math 4220M Miller G A 1935 Groups which are the products of two permutable proper subgroups Proceedings of the National Academy of Sciences 21 7 469 472 Bibcode 1935PNAS 21 469M doi 10 1073 pnas 21 7 469 PMC 1076628 PMID 16588002 Szep J 1950 On the structure of groups which can be represented as the product of two subgroups Acta Sci Math Szeged 12 57 61 Takeuchi M 1981 Matched pairs of groups and bismash products of Hopf algebras Comm Algebra 9 8 841 882 doi 10 1080 00927878108822621 Zappa G 1940 Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro Atti Secondo Congresso Un Mat Ital Bologna Edizioni Cremonense Rome 1942 119 125 Agore A L Chirvasitu A Ion B Militaru G 2007 Factorization problems for finite groups arXiv math 0703471 Bibcode 2007math 3471A doi 10 1007 s10468 009 9145 6 S2CID 18024087 Brin M G 2005 On the Zappa Szep Product Communications in Algebra 33 2 393 424 arXiv math 0406044 doi 10 1081 AGB 200047404 S2CID 15169734 Retrieved from https en wikipedia org w index php title Zappa Szep product amp oldid 1068695101, wikipedia, wiki, book, books, library,
