Central product
Encyclopedia
In mathematics
, especially in the field of group theory
, the central product is way of producing a group
from two smaller groups. The central product is similar to the direct product
, but in the central product two isomorphic
central
subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.
, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled.
A group G is an internal central product of two normal subgroups H, K if H ∩ K is contained in the center of G, . Sometimes the stricter requirement that H ∩ K is exactly equal to the center is imposed, as in .
The external central product is constructed from two groups H and K,
two subgroups H1 ≤ Z(H), K1 ≤ Z(K), and a group isomorphism θ:H1 → K1. The external central product is the quotient of the direct product H × K by the normal subgroup N = { ( h, k ) : where h in H1, k in K1, and θ(h)⋅k = 1 }, . Sometimes the stricter requirement that H1 = Z(H) and K1 = Z(K) is imposed, as in .
An internal central product is isomorphic to an external central product with H1 = K1 = H ∩ K and θ the identity. An external central product is an internal central product of the images of H × 1 and 1 × K in the quotient group ( H × K ) / N. This is shown for each definition in and .
Note that the external central product is not in general determined by its factors H and K alone. The isomorphism type of the central product will depend on the isomorphism θ. It is however well defined in some notable situations, for example when H and K are both finite extra special group
s and H1 = Z(H) and K1 = Z(K).
of central products is very similar to the representation theory of direct products, and so is well understood, .
Central products occur in many structural lemmas, such as which is used in George Glauberman
's result that finite groups admitting a Klein four group of fixed-point-free automorphisms are solvable
.
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 in the field of 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 central product is way of producing 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...
from two smaller groups. The central product is similar to the direct product
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...
, but in the central product two isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
central
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups.
Definition
There are several related but distinct notions of central product. Similarly to the direct productDirect 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...
, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled.
A group G is an internal central product of two normal subgroups H, K if H ∩ K is contained in the center of G, . Sometimes the stricter requirement that H ∩ K is exactly equal to the center is imposed, as in .
The external central product is constructed from two groups H and K,
two subgroups H1 ≤ Z(H), K1 ≤ Z(K), and a group isomorphism θ:H1 → K1. The external central product is the quotient of the direct product H × K by the normal subgroup N = { ( h, k ) : where h in H1, k in K1, and θ(h)⋅k = 1 }, . Sometimes the stricter requirement that H1 = Z(H) and K1 = Z(K) is imposed, as in .
An internal central product is isomorphic to an external central product with H1 = K1 = H ∩ K and θ the identity. An external central product is an internal central product of the images of H × 1 and 1 × K in the quotient group ( H × K ) / N. This is shown for each definition in and .
Note that the external central product is not in general determined by its factors H and K alone. The isomorphism type of the central product will depend on the isomorphism θ. It is however well defined in some notable situations, for example when H and K are both finite extra special group
Extra special group
In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p1+2n. Extra special groups often occur in centralizers of...
s and H1 = Z(H) and K1 = Z(K).
Examples
- Every extra special groupExtra special groupIn group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p1+2n. Extra special groups often occur in centralizers of...
is a central product of extra special groups of order p3. - The layer of a finite group, that is, the subgroup generated by all subnormalSubnormal subgroupIn mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....
quasisimple subgroupsQuasisimple groupIn mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S...
, is a central product of quasisimple groups in the sense of Gorenstein.
Applications
The representation theoryRepresentation theory of finite groups
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction...
of central products is very similar to the representation theory of direct products, and so is well understood, .
Central products occur in many structural lemmas, such as which is used in George Glauberman
George Glauberman
George Glauberman is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem....
's result that finite groups admitting a Klein four group of fixed-point-free automorphisms are solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
.