Review of Group Theory: Semidirect Products (Pt. II)
Point of post: This is a continuation of this post.
It’s interesting then to see exactly what the semidirect product “means” in terms of subgroups. By this, I mean that the direct product of groups can be thought of as just the internal product of subgroups, but there was some cosmic accident and they actually aren’t subsets of any coherent group. The question then remains how we can interpret this for semidirect products. We begin by noticing something interesting about certain subgroups of and use this to characterize the semidirect product
Theorem: Let and be groups and . Then, if and are defined as before then for all :
This is evidently surjective and a homomorphism. Indeed:
Noting though that
enables to conclude, from the First Isomorphism Theorem, that and .
: This follows from the fact that if then
: This is immediate.
: Let be arbitrary. Then, We merely note that
from where the result is clear.
: This is just a simple computation:
In fact, the first four of the above conditions characterize the semidirect product in the following sense:
Theorem: Let be a group with and . Moreover, if and then where
where is a variant of the inner automorphism . (note that this makes sense that since is normal and so ).
Proof: It’s evident from the conditions that and that every element of may be writtenuniquely as the product with and . Indeed, by the fact that we have that every element of may be written as the product of and . Note though that if then and thus so that or from where it quickly follows that and thus uniqueness is established. Thus, the mapping
is a bijection. Note though that
from where it follows that is a homomorphism and thus as desired.
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2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200
3.Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.