semidirect product of groups
The goal of this exposition is to carefully explain the correspondence between the notions of external and internal semi–direct products of groups, as well as the connection between semi–direct products and short exact sequences.
Naturally, we start with the construction of semi–direct products.
Definition 1.
Let and be groups and let be a group homomorphism. The semi–direct product is defined to be the group with underlying set and group operation .
We leave it to the reader to check that is really a group. It helps to know that the inverse of is .
For the remainder of this article, we omit from the notation whenever this map is clear from the context.
Set . There exist canonical monomorphisms and , given by
where (resp. ) is the identity element of (resp. ). These monomorphisms are so natural that we will treat and as subgroups of under these inclusions.
Theorem 2.
Let as above. Then:
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is a normal subgroup of .
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.
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Proof.
Let be the projection map defined by . Then is a homomorphism with kernel . Therefore is a normal subgroup of .
Every can be written as . Therefore .
Finally, it is evident that is the only element of that is of the form for and for . ∎
This result motivates the definition of internal semi–direct products.
Definition 3.
Let be a group with subgroups and . We say is the internal semi–direct product of and if:
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is a normal subgroup of .
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We know an external semi–direct product is an internal semi–direct product (Theorem 2). Now we prove a converse (Theorem 5), namely, that an internal semi–direct product is an external semi–direct product.
Lemma 4.
Let be a group with subgroups and . Suppose and . Then every element of can be written uniquely in the form , for and .
Proof.
Since , we know that can be written as . Suppose it can also be written as . Then so . Therefore and . ∎
Theorem 5.
Suppose is a group with subgroups and , and is the internal semi–direct product of and . Then where is given by
Proof.
Consider the external semi–direct product with subgroups and . We know from Theorem 5 that is isomorphic to the external semi–direct product , where we are temporarily writing for the conjugation map of Theorem 5. But in fact the two maps and are the same:
In summary, one may use Theorems 2 and 5 to pass freely between the notions of internal semi–direct product and external semi–direct product.
Finally, we discuss the correspondence between semi–direct products and split exact sequences of groups.
Definition 6.
An exact sequence of groups
is split if there exists a homomorphism such that is the identity map on .
Theorem 7.
Let , , and be groups. Then is isomorphic to a semi–direct product if and only if there exists a split exact sequence
Proof.
First suppose . Let be the inclusion map and let be the projection map . Let the splitting map be the inclusion map . Then the sequence above is clearly split exact.
Now suppose we have the split exact sequence above. Let be the splitting map. Then:
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, so is normal in .
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For any , set . Then , so . Set . Then . Therefore .
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Suppose is in both and . Write . Then , so . Therefore , so .
This proves that is the internal semi–direct product of and . These are isomorphic to and , respectively. Therefore is isomorphic to a semi–direct product . ∎
Thus, not all normal subgroups give rise to an (internal) semi–direct product . More specifically, if is a normal subgroup of , we have the canonical exact sequence
We see that can be decomposed into as an internal semi–direct product if and only if the canonical exact sequence splits.
Title | semidirect product of groups |
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Canonical name | SemidirectProductOfGroups |
Date of creation | 2013-03-22 12:34:49 |
Last modified on | 2013-03-22 12:34:49 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 10 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 20E22 |
Synonym | semidirect product |
Synonym | semi-direct product |