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 ( ) and F ( .) 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:
The multiplication of two elements of G is defined as
(I don't know what is standard notation here. Wikipedia would write it as
I have chosen
as where A and B are two automorphisms on H:
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, is not the identity in G.
I would have expected to be the identity for G.
Second, what I appeared to find as the identity, , is only the identity for part of the group. We have
So 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?