Section 7.2 Semidirect Products
Subsection External Semidirect Products
βDonβt give up and always keep on believing in your product.ββNiels van Deuren
We now discuss an important generalization for the direct product and a new method of constructing a new groups from the action of one group on another.
Suppose is a group with subgroups and such that Then we still have letβs see what we would need the multiplication on the cartesian product to be in order for defined by to still be a group homomorphism:
This means that we would need to have for to be a homomorphism. This motivates the following definition.
Discussion 7.1.
Decide amongst yourselves whether it should be spelled "semidirect", "semi-direct", "semi direct".
Before we prove that the construction above actually gives a group, letβs compute a few examples.
Example 7.12. Semidirect, Sans the Semi.
Example 7.13. .
Example 7.14. .
Let be cyclic of order and be cyclic of order for any There is an automorphism of that sends to This automorphism is its own inverse; i.e., it has order Therefore, by Proposition 2.28, there is a group homomorphism
with We may thus form the group
Then
and
Looks familar!
Indeed, by the universal mapping property for we have a homomorphism
and since it follows that is a bijection. So the dihedral group is a semidirect product, in which the two component groups are cyclic of orders and respectively:
and is the inversion homomorphism as described above.
Theorem 7.15. Semidirect Products are Groups.
is a group and
Proof.
-
The proof is straightforward but a bit messy. For associativity, note thatOn the other handThis gives associativity.The fact that
is a two-sided identity follows from the fact thatFinallyand similarly -
Define a funtionas
Then is a homomorphism, sinceThe map is clearly injective and hence its image is isomorphic to In fact, the image is normal since the second component ofis clearly Let us write this image asThe functiondefined by is also an injective homomorphism and thus its imageis isomorphic to is typically not normal, however. Finally, it is easy to see that and Putting this all together we have and
- Consider the projection onto the second factor
given by This is a goup homomorphism since the second component of is and is surjective by definition. NowBy the First Isomorphism Theorem for Groups we conclude that
Subsection Internal Semidirect Products
βIβve often said thereβs nothing better for the inside of a man than the outside of a horse.ββRonald Reagan
We can turn this around.
β1β
the semidirect product, not the Reagan quote.
Theorem 7.16. Recognition Theorem for Internal Semidirect Products.
Let be the permutation representation of the action of on via automorphisms given by Conjugation in (This means that for any where is the function for all ) Then the function
defined by is an isomorphism of groups. Moreover, under this isomorphism, corresponds to and corresponds to (referring to the notation in Theorem 7.15 above).
Proof.
We have
and thus is a homomorphism. Itβs kernel is which is just since The image of is clearly This proves is an isomorphism. It is obvious that and
Definition 7.17. Internal Semidirect Products.
Example 7.18. .
Example 7.19. .
It is important to be aware that for a fixed pair of groups and different actions of on via automorphisms can result in isomorphic semi-direct products. Indeed, determining when is in general a tricky business. The previous example shows this.
Example 7.20. .
and
Yet
since each is isomorphic to
On HW 8 you will give a more conceptual reason for why these two semidirect products turned out to be isomorphic: it is because and are conjugate in More generally, below is a criterion for a two semidirect products to be isomorphic.
Theorem 7.21. Isomorphisms of Semidirect Products.
Let be a finite cyclic group and let be an arbitrary group. Suppose that the images of and are conjugate subgroups of Then
Subsection Groups of Order
βI did not know I was in my prime until afterwards.ββMason Cooley
Theorem 7.22. Groups of Order .
Proof.
Let be a group of order Cayleyβs Theorem gives that there exist elements with and Let and Since is a normal subgroup of and since is a common subgroup of and Lagrangeβs Theorem gives that Thus and since we deduce that Theorem 7.16 now gives that is the internal semidirect product of and More to the point, where gives the action of on by Conjugation.
We now analyze the possibilities for By Proposition 2.23, There are two possibilities for the image of either or
If then (which implies ) and Therefore, in this case where the last isomorphism uses the Sunziβs Remainder Theorem.
If then is the map and by an earlier example for this we have so
Finally, because the former is abelian and the latter is not.
Letβs repeat the previous example for classifying groups of order with distinct primes into isomorphism classes.
Theorem 7.23. Classification for Groups of Order .
Let be primes.
- If
there is a unique group of order up to isomorphism, namely - If
there are exactly two groups of order up to isomorphism, namely and a non-abelian group.
Proof.
Let be a group of order and let be Sylow subgroups of order and respectively. We see that is a normal subgroup using Theorem 5.16, since is the smallest prime that divides
Furthermore, since is a subgroup of both and we have by Lagrangeβs Theorem that so that From here it follows that
and so Theorem 7.16 now yields that
for some homomorphism equivalently By Proposition 2.28 to give such a homomorphism is equivalent to giving an element so that which will give for Thus yielding that either or
Case 1: if then is the trivial homomorphism and thus
Case 2: if then it must be the case by Lagrange that By Corollary 2.24 we know that is a cyclic group. Therefore we have that if and only if there exists an element of order by Theorem 2.26 (2). Moreover any such element generates a subgroup of of order Since there is a unique subgroup of a cyclic group of a given order by Theorem 2.26 (2) we see that the image of is independent of the choice of Thus by Theorem 7.21 we conclude that all subgroups resulting from any choice of of order are isomorphic.
Moreover, from the explicit presentation of semidirect products of cyclic groups given in a homework problem we see that the resulting group is non-abelian; in particular it is not isomorphic to
Summary
- External Semidirect Products are defined using homomorphisms into automorphism groups.They are often used to construct non-abelian groups.
β2β
See: Theorem 7.15 - Internal Semidirect Products are similar to their direct counterparts, though only require that one of
or be normal. They can be identified using the Recognition Theorem for Internal Semidirect Products. - Using semidirect products, we are able to provide a Classification for Groups of Order
.