# Semidirect Product Page 5 - Honors Brown

• Artusartos
In summary, the internal and external versions of the semidirect product may seem the same, but they have slight differences in terms of their definitions. The internal version requires one subgroup to be a subgroup of the other, while the external version does not have this requirement. Additionally, the external version allows for the specification of an action between the two subgroups, while the internal version does not mention ordered pairs or actions. However, both versions can be used to show that any two groups are isomorphic to the pair of subgroups occurring in an internal product.
Artusartos
Page 5 in this link:

http://doctorh.umwblogs.org/files/2010/10/honors_brown.pdf

I couldn't really understand the difference between the internal version and the external version of the semidirect product...don't they both say the same thing?

$$(h_1, n_1)(h_2,n_2) = (h_1h_2, \phi_{h_2^{-1}}(n_1)n_2)$$?

Artusartos said:
Page 5 in this link:

http://doctorh.umwblogs.org/files/2010/10/honors_brown.pdf

I couldn't really understand the difference between the internal version and the external version of the semidirect product...don't they both say the same thing?

$$(h_1, n_1)(h_2,n_2) = (h_1h_2, \phi_{h_2^{-1}}(n_1)n_2)$$?

In that paper, the definition of the internal version does not involve "ordered pairs of elements" from two groups that have possibiley different group operations. The internal version requires that N be a subgroup of G. The external version does not. In the internal version, $n_1 g_1$ has an unambiguous interpretation and no mention is made of ordered pairs.

Stephen Tashi said:
In that paper, the definition of the internal version does not involve "ordered pairs of elements" from two groups that have possibiley different group operations. The internal version requires that N be a subgroup of G. The external version does not. In the internal version, $n_1 g_1$ has an unambiguous interpretation and no mention is made of ordered pairs.

Thanks a lot :)

i may not be able to add anything, this is sort of hard to describe. in the internal version, you are already given the big group G and two of its subgroups A,B, and you are saying that all elements of the big group are obtained in a certain way from elements of the two subgroups.

in the external version you are given just two distinct groups A' and B', not subgroups of anything else, and you ask whether there is some big group G which has subgroups A,B which are isomorphic to A' and B', and such that G is obtained from A and B as above.

i.e.given two groups A' and B', you can ask whether there exists a group G which is the internal direct product of two subgroups A,B isomorphic to A' and B'. The answer is yes, because the external product construction let's you build one out of the set theoretic product A'xB'. In this construction, the subgroup A = A'x{e} is isomorphic to A', for example.Actually to make the semi - direct construction you need a little more data, namely an action of one group on the other, I believe.So in a sense they really are the same, but the possibility of making the external product construction answers the question whether every pair of groups is isomorphic to the pair of subgroups occurring in an internal product.

I.e. theorem: given any two groups A',B', there exists a group G and subgroups A,B isomorphic to A' and B', such that G is an internal product of A and B.
proof: let G be the external product of A' and B', and take A = A'x{e} and B = {e}xB'.in fact i believe the construction also let's you specify the action of one subgroup on the other arbitrarily, as well as decide which one you want to be normal.

more precisely:

Theorem: Let H',K' be groups and let a'-->Aut(K') be a homomorphism. then there exists a group G which is an internal semi direct product of two subgroups H,K isomorphic to H' and K', such that K is normal in G, and H acts by conjugation on K according to the action corresponding to the homomorphism a.

Last edited:
mathwonk said:
i may not be able to add anything, this is sort of hard to describe. in the internal version, you are already given the big group G and two of its subgroups A,B, and you are saying that all elements of the big group are obtained in a certain way from elements of the two subgroups.

in the external version you are given just two distinct groups A' and B', not subgroups of anything else, and you ask whether there is some big group G which has subgroups A,B which are isomorphic to A' and B', and such that G is obtained from A and B as above.

i.e.given two groups A' and B', you can ask whether there exists a group G which is the internal direct product of two subgroups A,B isomorphic to A' and B'. The answer is yes, because the external product construction let's you build one out of the set theoretic product A'xB'. In this construction, the subgroup A = A'x{e} is isomorphic to A', for example.

Actually to make the semi - direct construction you need a little more data, namely an action of one group on the other, I believe.

So in a sense they really are the same, but the possibility of making the external product construction answers the question whether every pair of groups is isomorphic to the pair of subgroups occurring in an internal product.

I.e. theorem: given any two groups A',B', there exists a group G and subgroups A,B isomorphic to A' and B', such that G is an internal product of A and B.
proof: let G be the external product of A' and B', and take A = A'x{e} and B = {e}xB'.

in fact i believe the construction also let's you specify the action of one subgroup on the other arbitrarily, as well as decide which one you want to be normal.

more precisely:

Theorem: Let H',K' be groups and let a'-->Aut(K') be a homomorphism. then there exists a group G which is an internal semi direct product of two subgroups H,K isomorphic to H' and K', such that K is normal in G, and H acts by conjugation on K according to the action corresponding to the homomorphism a.

Thanks a lot

## What is a semidirect product?

A semidirect product is a mathematical concept that combines two groups together in a specific way. It is denoted by G = N ⋊ H, where N and H are two groups and ⋊ represents a special operation called the semidirect product.

## How is a semidirect product different from a direct product?

The main difference between a semidirect product and a direct product is that in a semidirect product, the two groups N and H are not required to be normal subgroups of G. This allows for a more general and flexible structure compared to the direct product.

## What is the significance of the semidirect product in group theory?

The semidirect product is important in group theory because it allows us to construct new groups from existing ones. This is particularly useful in understanding the structure and properties of larger, more complex groups by breaking them down into simpler components.

## What are some applications of semidirect products?

Semidirect products have many applications in mathematics and other fields such as physics and computer science. They are used in the study of symmetry in crystals, coding theory, and the theory of Lie algebras, among others.

## Can the semidirect product be extended to more than two groups?

Yes, the concept of semidirect product can be extended to any number of groups. This is known as the generalized semidirect product, and it allows for even more flexibility in constructing and studying new groups.

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