Wreath product

In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.

The "standard" or "unrestricted" wreath product of a group "A" by a group "H" is written as "A" _wr "H", or also "A" ≀ "H". In addition, a more general version of the product can be defined for a group "A" and a transitive permutation group "H" acting on a set "U", written as "A" _wr ("H", "U"). By Cayley's theorem, every group "H" is a transitive permutation group when acting on itself; therefore, the former case is a particular example of the latter.

An important distinction between the wreath product of groups "A" and "H", and other products such as the direct sum, is that the actual product is a semidirect product of "multiple copies" of "A" by "H", where "H" acts to permute the copies of "A" among themselves.

Definition

Our first example is the wreath product of a group "A" and a finite group "H". By Cayley's theorem, we may regard "H" as a subgroup of the symmetric group S_"n" for some positive integer "n".

We start with the set "G" = "A" ^"n", which is the cartesian product of "n" copies of "A", each component "x"_"i" of an element "x" being indexed by [1,"n"] . We give this set a group structure by defining the group operation " · " as component-wise multiplication; i.e., for any elements "f", "g" in "G", ("f·g")_"i" = "f"_"i""g"_"i" for 1 ≤ "i" ≤ "n".

To specify the action "*" of an element "h" in "H" on an element "g" of "G" = "A"^"n", we let "h" permute the components of "g"; i.e. we define that for all 1 ≤ "i" ≤ "n", :("h"*"g")_"i" = "g"_{"h" ^-1("i")}

In this way, it can be seen that each "h" induces an automorphism of "G"; i.e., "h"*("f" · "g") = ("h"*"f") · ("h"*"g").

The unrestricted wreath product is a semidirect product of "G" by "H", defined by taking "A" _wr ("H", "n") as the set of all pairs { ("g","h") | "g" in "A"^"n", "h" in "H" } with the following rule for the group operation:

:( "f", "h" )( "g", "k" )=( "f" · ("h" * "g"), "hk")

More broadly, assume "H" to be any transitive permutation group on a set "U" (i.e., "H" is isomorphic to a subgroup of Sym("U")). In particular, "H" and "U" need "not" be finite. The construction starts with a set "G" = "A"^"U" of |"U"| copies of "A". (If "U" is infinite, we take "G" to be the external direct sum ∑_E { "A"_"u" } of |"U"| copies of "A", instead of the cartesian product). Pointwise multiplication is again defined as ("f" · "g")_"u" = "f"_"u""g"_"u" for all "u" in "U".

As before, define the action of "h" in "H" on "g" in "G" by

:("h" * "g")_"u" = "g"_{"h" ^-1("u")}

and then define "A" _wr ("H", "U") as the semidirect product of "A"^"U" by "H", with elements of the form ("g", "h") with "g" in "A"^"U", "h" in "H" and operation:

:( "f", "h" )( "g", "k" )=( "f" · ("h" * "g"), "hk")

just as with the previous wreath product.

Finally, since every group acts on itself transitively, we can take "U" = "H", and use the regular action of "H" on itself as the permutation group; then the action of "h" on "g" in "G" = "A"^"H" is

:("h" * "g")_"k" = "g"_{"h" ^-1"k"}

and then define "A" _wr "H" as the semidirect product of "A"^"H" by "H", with elements of the form ("g", "h") and again the operation:

:( "f", "h" )( "g", "k" )=( "f" · ("h" * "g"), "hk")

If "A" itself is a permutation group, then the wreath product "A"_wr"(H,U)" can also be given the structure of a permutation group in two standard ways. If "A" acts on "X", then the two actions of the wreath product are:
* The imprimitive wreath product action on "X"×"U". If an element "a" of "A" takes "x" to "y" in its action on "X", then when "a" is considered as an element in the "u"th component of "A"^"U" then it acts as the identity on all ("x","v") with "v"≠"u" and takes ("x","u") to ("y","u"). The action of and element of "H" is similar, if "h" takes "u" to "v", then in the wreath product action it takes ("x","u") to ("x","v"). In other words, the set "X"×"U" is partitioned into blocks "X"×{"u"}, and the action of the "u"th copy of "A" is the same as the normal action of "A" on "X", the action of the "v"th copy of "A" is trivial when "v"≠"u", and the action of "H" is merely to permute the blocks. Thus ("f","h")("x","u")=("f"_"hu"("x"),"h"("u")).
* The primitive wreath product action on "X"^"U". The elements of the set "X"^"U" can be viewed as vectors with components indexed by "u". An element "a" of "A" when considered to be in the "u"th component of "A"^"U" acts on the vector by acting as the identity on all but the "u"th component where it acts normally as "a" in "A" on "X". The elements of "H" simply permute the components of the vectors.

Examples

* A nice example to work out is $mathbb\{Z\}wr\; Z\_3$, where $Z\_3$ is the cyclic group of order three.

* $Z\_2\; wr\; (S\_n,\; n)$is isomorphic to the group of signed permutation matrices of degree "n".

* $S\_n\; wr\; Z\_2$ is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.

* The Sylow p-subgroup of the symmetric group on "p"^"2" points is the wreath product $Z\_p\; wr\; Z\_p$. The Sylow p-subgroup of the symmetric group on "p"^"n+1" points is the wreath product of "Z"_"p" with the Sylow p-subgroup of the symmetric group on "p"^"n" points, sometimes called the ("n"+1)-fold iterated wreath product of "Z"_"p". More generally, the Sylow p-subgroup of any symmetric group on finitely many points is a direct product of iterated wreath products of "Z"_"p".

* Similarly, every maximal p-subgroup of the general linear group $operatorname\{GL\}(d,\{Bbb\; Z\})$ is conjugate in $operatorname\{GL\}(d,\{Bbb\; Q\})$ to a particular direct product of iterated wreath products of "Z"_"p".

Properties

* Every extension of "A" by "H" is isomorphic to a subgroup of $A\; wr\; H$.

* The elements of $A\; wr\; H$ are often written ("g","h") or even "gh" (with "g" in "A"^"H"). First note that ("e", "h")("g", "e") = ("g", "h"), and ("g", "e")("e", "h") = (("h"*"g"), "h"). So ("h" ^-1*"g", "e")("e", "h") = ("g", "h"). Consider both "G" = "A"^"H" and "H" as actual subgroups of $A\; wr\; H$ by taking "g" for ("g", "e") and "h" for ("e", "h"). Then for all "g" in "A"^"H" and "h" in "H", we have that "hg" = ("h" ^-1*"g")"h".

* The product ("g","h")("g' ","h' ") is then easier to compute if we write ("g","h")("g' ","h' ") as "ghg'h' " and push "g' " to the left using the commutative rule: :"h" {"g' "_"k"} = {"g' "_"hk"} "h" for all "k" in "H"so that :"ghg'h' " = {"g"_"k""g' "_"hk"}"hh' " for all "k" in "H".

* The wreath product is associative in that there is a natural isomorphism between $(G\; wr\; H)\; wr\; K$ and $G\; wr\; (H\; wr\; K)$. Indeed, this isomorphism is an isomorphism of permutation representations when using the imprimitive action and is effected by the natural set isomorphism from ("X" × "Y") × "Z" to "X" × ("Y" × "Z"). A natural isomorphism of permutation representations for a mixture of imprimitive and product actions is effected by the natural set isomorphism from ("X"^"Y")^"Z" to "X"^{("Y" × "Z")}.

* The wreath product is not in general commutative, in that for most groups $G\; wr\; H$ is not isomorphic to $H\; wr\; G$. Indeed, when "G" and "H" are finite these groups do not even usually have the same number of elements.
* Every imprimitive permutation group "G" naturally defines a partition of the set into blocks. If "A" is the stabilizer of one of the blocks "X", then the quotient group of "G" by the normal core of "A" can be identified as a transitive permutation group ("H","U") on the set of blocks, "U". "A" itself is a permutation group acting on the single block "X". Using the blocks to identify the domain of "G" with "X"×"U", there is a natural embedding of "G" into the imprimitive wreath product $(A,X)\; wr\; (H,U)$.

External links

* [http://planetmath.org/encyclopedia/WreathProduct.html PlanetMath page]