The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of

group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

first published in 1951 by Marc Krasner
Marc Krasner (1912 – 13 May 1985, in Paris) was a Russian Empire-born French mathematician, who worked on algebraic number theory.
Krasner emigrated from the Soviet Union to France and received in 1935 his PhD from the University of Paris under ...

and Lev Kaluznin. The theorem states that any group extension of a group by a group is isomorphic to a subgroup of the regular wreath product
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...

The theorem is named for the fact that the group is said to be '' universal'' with respect to all extensions of by
Statement

Let and be groups, let be the set of all functions from to and consider theaction
ACTION is a bus operator in Canberra, Australia owned by the ACT Government.
History
On 19 July 1926, the Federal Capital Commission commenced operating public bus services between Eastlake (now Kingston, Australian Capital Territory, Ki ...

of on itself by right multiplication. This action extends naturally to an action of on defined by $\backslash phi(g).h=\backslash phi(gh^),$ where $\backslash phi\backslash in\; K,$ and and are both in This is an automorphism of so we can define the semidirect product called the ''regular wreath product'', and denoted or $A\backslash wr\; H.$ The group (which is isomorphic to $\backslash $) 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
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

and then there is an injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

of groups $\backslash theta:G\backslash to\; A\backslash wr\; H$ such that maps onto $\backslash text(\backslash theta)\backslash cap\; K.$ 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 $\backslash psi:G\backslash to\; H$ whose kernel is Choose a set $T=\backslash $ of (right) coset representatives of in where $\backslash psi(t\_u)=u.$ Then for all in $t\_u\; x\; t^\_\backslash in\backslash ker\; \backslash psi=A.$ For each in we define a function such that $f\_x(u)=t\_u\; x\; t^\_.$ Then the embedding $\backslash theta$ is given by $\backslash theta(x)=(f\_x,\backslash psi(x))\backslash in\; A\backslash wr\; H.$ We now prove that this is a homomorphism. If and are in then $\backslash theta(x)\backslash theta(y)=(f\_x(f\_y.\backslash psi(x)^),\backslash psi(xy)).$ Now $f\_y(u).\backslash psi(x)^=f\_y(u\backslash psi(x)),$ so for all in :$f\_x(u)(f\_y(u).\backslash psi(x))\; =\; t\_u\; x\; t^\_\; t\_\; y\; t^\_=t\_u\; xy\; t^\_,$ so Hence $\backslash theta$ is a homomorphism as required. The homomorphism is injective. If $\backslash theta(x)=\backslash theta(y),$ then both (for all ''u'') and $\backslash psi(x)=\backslash psi(y).$ Then $t\_u\; x\; t^\_=t\_u\; y\; t^\_,$ but we can cancel and $t^\_=t^\_$ from both sides, so hence $\backslash theta$ is injective. Finally, $\backslash theta(x)\backslash in\; K$ precisely when $\backslash psi(x)=1,$ in other words when $x\backslash in\; A$ (as $A=\backslash ker\backslash psi$).Generalizations and related results

*The Krohn–Rhodes theorem is a statement similar to the universal embedding theorem, but forsemigroup
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

s. A semigroup is a ''divisor'' of a semigroup if it is the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of a subsemigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', ...

of under a homomorphism. The theorem states that every finite semigroup is a divisor of a finite alternating wreath product of finite simple group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 ''x'' ∈ ''S'' is aperiodic, that is, for each ''x'' there exists a positive integer ''n'' such that ''x'n'' = ''x'n'' + 1. An aperiodic monoid is an aperiod ...

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