Universal embedding theorem
   HOME

TheInfoList



OR:

The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of
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 ( ...
first published in 1951 by Marc Krasner and Lev Kaluznin. The theorem states that any
group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...
of a group by a group is isomorphic to a subgroup of the regular
wreath product In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used ...
The theorem is named for the fact that the group is said to be ''
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company that is a subsidiary of Comcast ** Universal Animation Studios, an American Animation studio, and a subsidiary of N ...
'' with respect to all extensions of by


Statement

Let and be groups, let be the set of all functions from to and consider the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of on itself by multiplication. This action extends naturally to an action of on , defined as (h\cdot \phi)(g)=\phi(h^g), where \phi\in K, and and are both in This is an automorphism of so we can construct the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
, which is termed the ''regular
wreath product In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used ...
'', and denoted or A\wr H. The group (which is isomorphic to \) is called the ''base group'' of the wreath product. The Krasner–Kaloujnine universal embedding theorem states that if has a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
and then there is an
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of groups \theta:G\to A\wr H such that maps surjectively onto \text(\theta)\cap K. This is equivalent to the wreath product having a subgroup isomorphic to where is any extension of by


Proof

This proof comes from Dixon–Mortimer.. Define a homomorphism \psi:G\to H whose kernel is Choose a set T=\ of (right) coset representatives of in where \psi(t_u)=u. Then for all in t_u^ x t_\in\ker \psi=A. For each in we define a function f_x:H\to A such that f_x(u)=t_u^ x t_. Then the embedding \theta is given by \theta(x)=(f_x,\psi(x))\in A\wr H. We now prove that this is a homomorphism. If and are in then \theta(x)\theta(y)=(f_x(\psi(x).f_y),\psi(xy)). Now \psi(x).f_y(u)=f_y(\psi(x)^u), so for all in :f_x(u)(\psi(x).f_y(u)) = t_u^ x t_ t_^ y t_=t_u xy t^_, so Hence \theta is a homomorphism as required. The homomorphism is injective. If \theta(x)=\theta(y), then both (for all ''u'') and \psi(x)=\psi(y). Then t_u^ x t_=t_u^ y t_, but we can cancel t^_ and t_=t_ from both sides, so hence \theta is injective. Finally, \theta(x)\in K precisely when \psi(x)=1, in other words when x\in A (as A=\ker\psi).


Generalizations and related results

*The Krohn–Rhodes theorem is a statement similar to the universal embedding theorem, but for
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s. A semigroup is a ''divisor'' of a semigroup if it is the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of a
subsemigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the ...
of under a homomorphism. The theorem states that every finite semigroup is a divisor of a finite alternating wreath product of finite
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s (each of which is a divisor of ) and finite
aperiodic semigroup In mathematics, an aperiodic semigroup is a semigroup ''S'' such that every element is aperiodic, that is, for each ''x'' in ''S'' there exists a positive integer ''n'' such that ''xn'' = ''x'n''+1. An aperiodic monoid is an aperiodic semigroup ...
s. *An alternate version of the theorem exists which requires only a group and a subgroup (not necessarily normal).. In this case, is isomorphic to a subgroup of the regular wreath product


References


Bibliography

* * * *{{cite book , last1=Praeger , first1=Cheryl , last2=Schneider , first2=Csaba , title=Permutation groups and Cartesian Decompositions , date=2018 , publisher=Cambridge University Press , isbn=978-0521675062 , url=https://books.google.com/books?id=ISZaDwAAQBAJ , ref=PS Theorems in group theory