Universal embedding theorem

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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 the
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of on itself by right multiplication. This action extends naturally to an action of on defined by $\phi\left(g\right).h=\phi\left(gh^\right),$ where $\phi\in K,$ 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 $A\wr H.$ 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 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 $\theta:G\to A\wr H$ such that maps onto $\text\left(\theta\right)\cap K.$ 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 $\psi\left(t_u\right)=u.$ Then for all in $t_u x t^_\in\ker \psi=A.$ For each in we define a function such that $f_x\left(u\right)=t_u x t^_.$ Then the embedding $\theta$ is given by $\theta\left(x\right)=\left(f_x,\psi\left(x\right)\right)\in A\wr H.$ We now prove that this is a homomorphism. If and are in then $\theta\left(x\right)\theta\left(y\right)=\left(f_x\left(f_y.\psi\left(x\right)^\right),\psi\left(xy\right)\right).$ Now $f_y\left(u\right).\psi\left(x\right)^=f_y\left(u\psi\left(x\right)\right),$ so for all in :$f_x\left(u\right)\left(f_y\left(u\right).\psi\left(x\right)\right) = t_u x t^_ t_ y t^_=t_u xy t^_,$ so Hence $\theta$ is a homomorphism as required. The homomorphism is injective. If $\theta\left(x\right)=\theta\left(y\right),$ then both (for all ''u'') and $\psi\left(x\right)=\psi\left(y\right).$ Then $t_u x t^_=t_u y t^_,$ but we can cancel and $t^_=t^_$ from both sides, so hence $\theta$ is injective. Finally, $\theta\left(x\right)\in K$ precisely when $\psi\left(x\right)=1,$ 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 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
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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

# 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