We all know what a direct product of two groups A and B is. It is isomorphic to the group defined as follows : As a set it is and has the group operation where ac and bd make sense inside A and B respectively. And clearly you identify with A and with B and both these are normal subgroups inside .
To make a semidirect product of two groups A and B, we need more information. Namely, we need a homomorphism of which just means we need an action of B on A. This is what links the groups A and B together.
Then we define to be the semidirect product of A and B via f as follows :
- As a set, it is still
Notice that in the first coordinate, the map f is involved. A direct product is just where sends every . Thus and hence .
It is generally taught in any group theory course that you can find an isomorphic copy of A as a normal subgroup in and an isomorphic copy of B as a subgroup (not neccesarily normal) in .
Now here is a question :
Can a group be isomorphic to a nontrivial semidirect product of A and B (that is is not the trivial map) and also simultaneously be isomorphic to the direct product of A and B. In symbols, can you think of an example of A, B and f where
Note that we don’t demand that the isomorphism preserves A or B or anything like that.
OK, here’s an answer. Take A=B to be some nonabelian finite group and let . f is not the trivial map because there is some a, b such that because A=B is nonabelian.
- T is a group homomorphism because
- T is 1-1 because implies and and hence .
- Since the domain and codomain are finite and have the same size, T is onto and hence is a group isomorphism.
Here’s the “picture” behind the proof [I am not sure whether there really is a mathematical analogy, but this picture helped me formalize the example]. The real plane is the span of the x and the y-axis. It is also the span of the line y=x and the x-axis… With this as an intuition, can we try to symbolize everything into groups ?
Well, look at . It has as a normal subgroup. It has also the “diagonal” as a subgroup which is not normal. Clearly C and D are isomorphic to A and even more clearly . Also for any . Thus is the semidirect product (internal, if you will) of subgroups C and D ..