Zappa-Szep product
Encyclopedia
In mathematics
, especially group theory
, the Zappa–Szép product (also known as the knit product) describes a way in which a group
can be constructed from two subgroup
s. It is a generalization of the direct
and semidirect product
s. It is named after Guido Zappa
and Jenö Szép.
e, and let H and K be subgroups of G. The following statements are equivalent:
If either (and hence both) of these statements hold, then G is said to be an internal Zappa–Szép product of H and K.
of invertible n × n matrices
over the complex number
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 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 Hall's 1937 theorem on the existence of Sylow system
s 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, 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.
α : K × H → H and β : K × H → K which turn out to have the following properties:
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,
Then H × K is a group called the external Zappa–Szép product of the groups H and K. The subset
s H × {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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Zappa–Szép product (also known as the knit product) describes a way in which a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
can be constructed from two subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s. It is a generalization of the direct
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
and semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
s. It is named after Guido Zappa
Guido Zappa
Guido Zappa is an Italian mathematician and a noted group theorist: his other main research interests are geometry and complex analysis in one and several variables, and also the history of mathematics. Zappa is particularly known for some examples of algebraic curves that strongly influenced the...
and Jenö Szép.
Internal Zappa–Szép products
Let G be a group with identity elementIdentity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
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–Szép product of H and K.
Examples
Let G = GL(n,C), the general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
of invertible n × n matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s. For each matrix A in G, the QR decomposition
QR decomposition
In linear algebra, a QR decomposition of a matrix is a decomposition of a matrix A into a product A=QR of an orthogonal matrix Q and an upper triangular matrix R...
asserts that there exists a unique unitary matrix Q and a unique upper triangular matrix R with positive real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
entries on the main diagonal such that A = QR. Thus G is a Zappa–Szép product of the unitary group
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
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 system
Hall subgroup
In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They are named after the group theorist Philip Hall.- Definitions :A Hall divisor of an integer n is a divisor d of n such that...
s 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, 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 mappingsMap (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
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 bijectionBijectionA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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, h1), h2)
- 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
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
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–Szép product of the groups H and K. The subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
s H × {e} and {e} × K are subgroups isomorphic
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called 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 normalNormal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
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.