Need Help: Wreath Products & Standard Wreath Products

[happyg1]

Homework Statement

Hello,
My book has all of 1 page about wreath products and standard wreath products. I'm really lost on this topic and I have 4 homework problems using it. Does anyone know where I could find a little more info on the topic?

Homework Equations

The Attempt at a Solution

 

[Chris Hillman]

Intuition for wreath products

What book is that? Cameron, Permutation Groups is short and very readable and quite fascinating book, which might help. I don't seem to see the "relevant equations" (am I the only one?), but the basic idea is pretty simple: given two groups G,H acting on X,Y, to form the wreath product action you simply make |X| copies of the H-set Y and then let G act on that by permutating the copies as per the G-set X.

HTH

 

[happyg1]

This book is by Robinson "A Course in the Theory of Groups"
The "Relevant Equations" look like this

Let H and K be permutation groups acting on the sets X and Y respectively. Then H~K=<H(y),K^*|\y in Y>is the wreath product.

Then:
If H and K are arbitrary groups, we can think of them as permutation groups an their underlying sets via the right regular representation and form of their wreath product W=H~K: this is called the standard wreath product.

On of my HW problems is:
Prove that the standard wreath product Z~Z is finitely generated but that it has a non-finitely generated subgroup.

I'm just lost on this concept. I'll have to come back to it later. This book doesn't say anything about |X| copies. I'll try to find your book at the library.
CC

 

[NateTG]

This only deals with the finite case.
You're familiar with the direct product, right?

So, for example if we have a group H, then elements of
H \times H are ordered pairs (h_1,h_2) and the product is defined in the obvious way:
(h_1,h_2) \times (g_1,g_2)=(h_1 g_1, h_2 g_2)

And this readily generalizes to any finite power of H.

Now, let's say we have some finite group G with n=|G|. Then, we know that G &lt; S_n from Caley's theorem - that is, we can consider the elements of G to be permutations of n elements.

Now, consider H^n (which is the direct product of n copies of H). Each element of this group is an ordered n-tuple of elements of H.

If we consider the elements of G as permutations, we can allow them to act on the elements of H^n by permuting them:
If
\vec{h}=\left( h_1,h_2...h_n \right)
then
g \cdot \vec{h} = \left(h_{g(1)},h_{g(2)}...h_{g(n)}\right)

So, if we consider the elements of the cartesian product G \times H^n, we can define a group operation on them:
(g_1,\vec{h}_1) * (g_2,\vec{h}_2) = (g_1 g_2, g_1 \cdot (\vec{h}_1 \vec{h}_2))

 

[happyg1]

Wow! Thanks! That really helps! I am MUCH more clear on the concept now. I'll give my HW problem another look with your info at hand.
CC