As the work involved with helping me with this question might be rather extreme, I'll be satisfied with a pointer in the right direction.

I am trying to construct a group G as the semi-direct product of two groups H ($\displaystyle \{ e, a, b, c \}$) and F ( $\displaystyle \{ \epsilon, x \}$.) The multiplication tables for each are shown below in the attachment.

In order to do this I need to define the group elements for G. They are simply the direct product (taken in the sense of sets) of the groups F and H: $\displaystyle \{ (f, h)| \forall f \in F, h \in H \}$

The multiplication of two elements of G is defined as

$\displaystyle (f_1, h_1) ~ (f_2, h_2) = (f_1f_2, (h_1(f_2T))h_2)$

where $\displaystyle T \in Hom(F, Aut(H))$

(I don't know what is standard notation here. Wikipedia would write it as

$\displaystyle (f_1, h_1) ~ (f_2, h_2) = (f_1f_2, (T_{f_2}(h_1)h_2)$)

I have chosen

$\displaystyle T:~F \to Aut(H)$ as $\displaystyle T:~ \epsilon \to B, x \to A$ where A and B are two automorphisms on H:

$\displaystyle A:~H \to H$ as $\displaystyle A:~e \to a, a \to b, b \to c, c \to e$

$\displaystyle B:~H \to H$ as $\displaystyle B:~e \to c, a \to e, b \to a, c \to b$

The development of the multiplication table for G should be relatively simple: just apply the definition. But I am coming up with two strange facts before I even get very far.

First, $\displaystyle (\epsilon, e)$ is not the identity in G.

$\displaystyle (\epsilon, e)~(\epsilon, e) = (\epsilon \epsilon, (e( \epsilon T))e) = (\epsilon, (eB)e) = (\epsilon, c)$

I would have expected $\displaystyle (\epsilon, e)$ to be the identity for G.

Second, what I appeared to find as the identity, $\displaystyle (\epsilon, a)$, is only the identity for part of the group. We have

$\displaystyle (\epsilon , a)~(\epsilon , h) = (\epsilon, h)~\forall h \in H$

but

$\displaystyle (\epsilon, a)~(x, h) \neq (x, h)~\text{for any } h \in H$

So $\displaystyle (\epsilon, a)$ cannot be the identity either. In fact, though I admittedly haven't gone through the whole multiplication table, it would seem that G does not have an identity!

Obviously I'm doing something wrong. As I've never worked with semi-direct products before I'm guessing I somehow defined my function T wrong? Or is it something else?

Thanks!

-Dan