Page 107 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 107
mož Moravec: Some Topics in the Theory of Finite Groups 95
Another construction related to semidirect products is that of a wreath product. Let
G and H be groups and let H act on the set X = {x1, x2, . . . , xn }. We take
n
G X = Gxi
i =1
to be the direct product of n copies of G indexed by the set X . Then H also acts on G X by
the rule
(g x1 , g x2 , . . . , g xn )h = (g x1h , g x2h , . . . , g xn h ).
Therefore we have a homomorphism α: H → Aut(G X ) and we can form the semidirect
product H α G X which is denoted by G X H and called the wreath product of G by H .
A special case is when X = H , and H acts on X by right multiplication. Then the cor-
responding wreath product is denoted by G H and called the regular (standard) wreath
product. Here is an example of how to build C2 C4 with GAP:
gap> G := StandardWreathProduct(CyclicGroup(2), CyclicGroup(4));
gap> IdGroup(G);
[ 64, 32 ]
Alternatively, we can build C2 C4 as a semidirect product C4 C24, where we think of
4
C4 as the group 〈(1 2 3 4)〉 acting on C 2 by permuting the indices:
gap> G := SemidirectProduct(Group((1,2,3,4)), GF(2)^4);
gap> IdGroup(G);
[ 64, 32 ]
Wreath products are important in the theory of extensions because of the following:
Theorem 3.4.2 Every extension of G by H is isomorphic to a subgroup of G H .
We leave the proof as an exercise.
3.4.3 Extensions with abelian kernels
Consider
A / µ /E ε //G ,
where A is an abelian group (written additively). When choosing a transversal to M =
im µ = ker ε in E , we get a function τ: G → E defined by g τ = x , where x ∈ is such
that g = x ε (note that this is well defined). The function τ is called a transversal function.
Note that τ is not necessarily a homomorphism. We also see that τε = 1G , and that any
function τ: G → E with the property τε = 1G determines a transversal to M in E , namely
{g τ | g ∈ G }.
Another construction related to semidirect products is that of a wreath product. Let
G and H be groups and let H act on the set X = {x1, x2, . . . , xn }. We take
n
G X = Gxi
i =1
to be the direct product of n copies of G indexed by the set X . Then H also acts on G X by
the rule
(g x1 , g x2 , . . . , g xn )h = (g x1h , g x2h , . . . , g xn h ).
Therefore we have a homomorphism α: H → Aut(G X ) and we can form the semidirect
product H α G X which is denoted by G X H and called the wreath product of G by H .
A special case is when X = H , and H acts on X by right multiplication. Then the cor-
responding wreath product is denoted by G H and called the regular (standard) wreath
product. Here is an example of how to build C2 C4 with GAP:
gap> G := StandardWreathProduct(CyclicGroup(2), CyclicGroup(4));
gap> IdGroup(G);
[ 64, 32 ]
Alternatively, we can build C2 C4 as a semidirect product C4 C24, where we think of
4
C4 as the group 〈(1 2 3 4)〉 acting on C 2 by permuting the indices:
gap> G := SemidirectProduct(Group((1,2,3,4)), GF(2)^4);
gap> IdGroup(G);
[ 64, 32 ]
Wreath products are important in the theory of extensions because of the following:
Theorem 3.4.2 Every extension of G by H is isomorphic to a subgroup of G H .
We leave the proof as an exercise.
3.4.3 Extensions with abelian kernels
Consider
A / µ /E ε //G ,
where A is an abelian group (written additively). When choosing a transversal to M =
im µ = ker ε in E , we get a function τ: G → E defined by g τ = x , where x ∈ is such
that g = x ε (note that this is well defined). The function τ is called a transversal function.
Note that τ is not necessarily a homomorphism. We also see that τε = 1G , and that any
function τ: G → E with the property τε = 1G determines a transversal to M in E , namely
{g τ | g ∈ G }.
