Semidirect Product Page 5 - Honors Brown

[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?

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

Thanks in advance

 

[Stephen Tashi]

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.

 

[Artusartos]

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 :)

 

[mathwonk]

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.

 

[Artusartos]

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