External/internal direct product of groups

What is the internal and external direct product of groups? I understand that the external product of groups is the Cartesian product of the groups and where the binary operation is such that (a1,b1) ⊕(a2,b2) = (a1a2,b1b2) but I'm not sure about the internal direct product.

Reply 1

FireGarden

14

Original post by B_9710

What is the internal and external direct product of groups? I understand that the external product of groups is the Cartesian product of the groups and where the binary operation is such that (a1,b1) ⊕(a2,b2) = (a1a2,b1b2) but I'm not sure about the internal direct product.

I vaguely recall those, but really, unless it's specified for your course or something, I wouldn't worry about it: it literally makes no difference. It is a theorem that direct products of groups are unique up to unique isomorphism. The whole internal/external thing is going to be just two ways of looking at the same construction, but isn't really all that illuminating.

(edited 8 years ago)

Reply 2

Gregorius

14

Original post by B_9710

What is the internal and external direct product of groups? I understand that the external product of groups is the Cartesian product of the groups and where the binary operation is such that (a1,b1) ⊕(a2,b2) = (a1a2,b1b2) but I'm not sure about the internal direct product.

The motivation here is that If you have two groups

N_{1}

and

N_{2}

sitting on your desk, then you can form their external direct product

G

; you seem comfortable with that.

On the other hand, what happens if you have a group

G

wandering around the office and you wonder to yourself whether it is isomorphic to some external direct product of two groups? This is where the internal direct product comes in. If you can find two normal subgroups

N_{1}

and

N_{2}

that satisfy certain properties (their intersection is trivial and their product is

G

), then

G

is the internal direct product of

N_{1}

and

N_{2}

and

G

is isomorphic to the external direct product of two groups isomorphic to

N_{1}

and

N_{2}

.

So, basically, they are two different ways of looking at the same thing? Nearly. For the product of two groups, or for the product of a finite number of groups, the notions correspond. But if you were to start off with an infinite family of groups, then things become trickier and the correspondance between external and internal direct product doesn't go through in the same way. (It can be rescued, though, with some careful definition!)

Reply 3

B_9710

OP

18

Original post by Gregorius

The motivation here is that If you have two groups

N_{1}

and

N_{2}

sitting on your desk, then you can form their external direct product

G

; you seem comfortable with that.

On the other hand, what happens if you have a group

G

wandering around the office and you wonder to yourself whether it is isomorphic to some external direct product of two groups? This is where the internal direct product comes in. If you can find two normal subgroups

N_{1}

and

N_{2}

that satisfy certain properties (their intersection is trivial and their product is

G

), then

G

is the internal direct product of

N_{1}

and

N_{2}

and

G

is isomorphic to the external direct product of two groups isomorphic to

N_{1}

and

N_{2}

.

So, basically, they are two different ways of looking at the same thing? Nearly. For the product of two groups, or for the product of a finite number of groups, the notions correspond. But if you were to start off with an infinite family of groups, then things become trickier and the correspondance between external and internal direct product doesn't go through in the same way. (It can be rescued, though, with some careful definition!)