Zappa–Szép Product
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, 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
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
s. It is a generalization of the
direct Direct may refer to: Mathematics * Directed set, in order theory * Direct limit of (pre), sheaves * Direct sum of modules, a construction in abstract algebra which combines several vector spaces Computing * Direct access (disambiguation), ...
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).


Internal Zappa–Szép products

Let ''G'' be a group with
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
''e'', and let ''H'' and ''K'' be subgroups of ''G''. The following statements are equivalent: * ''G'' = ''HK'' and ''H'' ∩ ''K'' = * 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–Szép product of ''H'' and ''K''.


Examples

Let ''G'' = GL(''n'',C), the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
of
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
''n × n'' matrices over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. 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 In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek Î ...
such that ''A'' = ''QR''. Thus ''G'' is a Zappa–Szép 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–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 every alternating group of prime degree is also 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(''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''. From these, it follows that * For each ''k'' in ''K'', the mapping ''h'' α(''k'', ''h'') is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
of ''H''. * For each ''h'' in ''H'', the mapping ''k'' β(''k'', ''h'') is a bijection of ''K''. (Indeed, suppose α(''k'', ''h''1) = α(''k'', ''h''2). Then ''h''1 = α(''k''−1''k'', ''h''1) = α(''k''−1, α(''k'', ''h''1)) = α(''k''−1, α(''k'', ''h''2)) = ''h''2. 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'' → ''k''''h'' and the right action by ''k'' → ''k''''h'', then the last two properties amount to ''k''(''h''1''h''2) = ''k''''h''1 ''k''''h''1''h''2 and (''k''1''k''2)''h'' = ''k''1''k''2''h'' ''k''2''h''. 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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
''H'' × ''K'', define a multiplication and an inversion mapping by, respectively, * (''h''1, ''k''1) (''h''2, ''k''2) = (''h''1 α(''k''1, ''h''2), β(''k''1, ''h''2) ''k''2) * (''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
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s ''H'' × and × ''K'' are subgroups
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to ''H'' and ''K'', respectively, and ''H'' × ''K'' is, in fact, an internal Zappa–Szép product of ''H'' × and × ''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 (h_1k_1)(h_2k_2) = (h_1k_1h_2k_1^)(k_1k_2) and h_1k_1h_2k_1^\in H since by normality of H, k_1h_2k_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 α(''k'',''h'') = ''h''. In this case, ''G'' is an internal direct product of ''H'' and ''K''.


See also

Complement (group theory)


References

* , Kap. VI, §4. * . * * . * . * ; Edizioni Cremonense, Rome, (1942) 119–125. * . * {{DEFAULTSORT:Zappa-Szep product Group theory pl:Iloczyn kompleksowy