Split Exact Sequences and Semidirect Products
The direct product of two groups provides a special sort of short exact sequence. We know that there is a surjection , and we can build an injection from the identity homomorphism on and the homomorphism sending everything in to the identity of . Since the kernel of the latter is exactly the image of the former, this is an exact sequence .
We can do the same thing, swapping and , to get a sequence . If we compose the homomorphisms , we get exactly the identity homomorphism on . When we see this, we say the sequence splits.
Let’s look at this more generally. A split exact sequence is any short exact sequence where the surjection splits: . We can identify with its image in , which must be a normal subgroup since it’s also the kernel of the surjection onto . The homomorphism from to must also be an injection — if it had a nontrivial kernel it couldn’t be part of the identity homomorphism from to itself. We’ll identify with its image in as well.
Since is normal, it’s fixed under conjugation by any element of . In particular, conjugation by any element of sends elements of back into . This defines a homomorphism . We’ll usually write as The action of on an element of is given by in the group . It’s important to remember that we’re considering and as living inside the same group so that this conjugation makes sense.
Okay, so let’s turn this around and build the group up from the outside. We start with groups and , and a homomorphism . We can take the underlying set of to be all pairs with in and in . We define composition by . Verify for yourself that there is an identity and an inverse making this into a group. We call this group the “semidirect product” of and , and write , or just if the homomorphism is understood. If is the identity automorphism on for every , we just have the direct product back.
We can also write down generators and relations like we did for the direct product. If has generators and relations , while has generators and relations , the semidirect product has generators and relations . The elements of and don’t commute, but we can “pull past ” to the right by hitting with : .
One example of a semidirect product I think about is the group of “Euclidean motions” of the plane. We can slide figures around the plane without changing them, and we can turn them around some fixed origin point, but I don’t allow flipping them over. Sliding gives a group of translations and turning gives a group of rotations around the origin. Sliding and turning don’t commute: turning a triangle by 90° around the origin and moving it right an inch is different than moving it right an inch and turning it 90° around the origin. However, conjugating a translation by a rotation gives another translation, so the group of Euclidean motions is the semidirect product .
Another example that comes up is the wreath product, where is some subgroup of a permutation group , and is the direct product of copies of a group . We take the action of on to be permutations of the factors in the product: . This one will be useful to us soon.
One last note: as I was thinking about semidirect products in preparation for this post I was trying to determine if there is a universal property for them, like there is for the direct product. I asked a few other people too, and nobody seems to have a good answer. Since I know a few professionals are reading, does anyone know of a universal property that characterizes the semidirect product ?