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What is the difference between external and internal direct product ?? I think both of them boil down to the same thing.

user26857
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Iqra Altaf
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    I just try my best to forget the distinction even exists because I think it's unnecessary – Alex Mathers Oct 08 '15 at 07:35
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    I convinced myself like this: in an external direct product, the multiplication/operation works the way that it does by construction, whereas in internal you have a pre-existing operation that happens to let the original object be recovered from some subobjects/factors -- more like a decomposition or factorisation (vs. expansion), if you will. Other than the connotation, they really are the same; $K=G \cross H$ (where $G,H$ are given) makes the product feel external, but any equation can of course be read from right to left, and $G \cross H = K$ (where you start with $K$) seems internal. – Vandermonde Nov 15 '15 at 21:25
  • Need a small edit to your TEX command – Supriyo Halder Aug 09 '17 at 03:12

3 Answers3

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They are two different ways of looking at the same thing, but the definitions are basically equivalent.

Every internal direct product $G$ is naturally isomorphic to the external direct product (of its direct factors).

and

Every external direct product is naturally realized as an internal direct product.

The biggest distinction I've seen is that if $A,B \subset G $, and $A\times B \cong G$, we say $G$ is the internal direct product of $A$ and $B$. However, if $A,B$ are not subgroups of $G$ (rather, they are isomorphic to direct factors of $G$), we would say $G \cong A \times B$ is an external direct product.

fixedp
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Let $G$ be a group with identity element $e$. Let $H$ and $K$ be normal subgroups of $G$ such that $H \cap K= \{e\}$. Then $H \times K \simeq H \oplus K$ where $H \times K$ is the internal direct product between $H$ and $K$ and $H \oplus K$ is the external direct product between $H$ and $K$. When $H$ and $K$ have nontrivial intersection this may not hold.

Alex M.
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Nerdy
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  • If $H, K$ as normal and has the trivial intersection, can we say that any element of $H$ is commute with any element of $K$ ? – Bumblebee Jul 27 '17 at 00:34
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The difference is is that in the internal direct product is between the subgroups of the same group $G$ but in the external direct product it can be made for any two groups in which they don't have any relation between them

Abdo
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