Re: question about wreath product

Thank you for the reply, but I still don't understand some points.

We start with the group G I mentioned in the original post.

a b d e f g
\ / \ / \ /
\ / \ / \ /
c i j
G is the group of all order automoprhisms
of this partial ordered set.

This partial ordered set has three V, say V1, V2, V3.

Let G1 be the g in G such that g is the identity in V2 and V3.
G2 is the group of g in G such that g is the identity in V1 and V3. The
same for G3.

Now following Magidin's topics, I have:

1. G1, G2, G3 must intersect trivially, what is the case.
2. G1, G2, G3 must centralize each other what is the case clearly.
3. The elments of the group generated by G1, G2, and G3 are of the form
g=(g1, g2,g3) where
gi is an order automorphism of Vi.

( a b c d e i )
If g=( )
( b a c d e i .....)
then the conjugate of g by some h in G is

( h(a) h(b) h(c) h(d) h(e) h(i) )
h^-1gh=( )
( h(b) h(a) h(c) h(d) h(e) h(i) .....).

As h is in G, h(V1)=Vi; h(V2)=Vj, h(V3)=Vk. So h^-1gh is an element of
the form (g1,g2,g3) with gi in Gi. I think I have show that the group
generated by G1, G2, G3 is normal in G.

4. This group must be proper subgroup of G, and that is obvious because
in the group generated by G1,G2, G3 there is nothing such g(V1)=V2, but
there is one such g in G.

5. Now Magidin says it must have a complement C. I don't know what is a
complement. I can guess it is a group C such that G1, G2, G3 and C
generate G.

And here is my problem. In the definition of wreath product, we have
three copies of a group (in our case it is G1, G2, G3) and then we have
a subgroup of the symmetric group on three elements. So the complement
C is outside the group G. But if I understand well "complement",
Magidin says C must be inside the group G. But how can the group C then
permute the coordinates of G1xG2xG3 if the group is not permuting
{1,2,3}, but is a group of order automorphisms?

I don't understand also "you need H and kHk^{-1} to be either equal",
because C is a group of permutations on {1,2,3}. I cannot be sure that
kHk^{-1} exists. I can be multiplying here things that cannot be
multiplyed.

I presume this is a silly problem, but how can I solve it? Would you
please tell me how in the example of G above I find the complment C and
how the complment is a group of permutations of {1,2,3} inside G. Thank
you, thank you. Wreath product seems like being on the top of a wild
bull...

.