In the context of group theory, what does does the notation of

$$ G = G_1 \oplus G_2 \oplus \ldots \oplus G_n $$

typically denote? I'm assuming its distinct from the external direct product of groups $G_1 \times G_2 \times \ldots \times G_n$.

  • 2
    $\begingroup$ It' the direct sum, not the direct product. For a finite number of groups it's the same thing, but when you have infinitely many groups direct sum and direct product can look very different! $\endgroup$ – Ferra Apr 11 '14 at 13:28

The external direct sum $\bigoplus_{\alpha\in A}{G_\alpha}$ is defined to be the set of all tuples $(g_\alpha)_{\alpha\in A}$ where $g_\alpha\in G_\alpha$ with finite support (i.e. all but finitely-many of the entries are the identity of the respective group). Addition is componentwise, the identity is the tuple where every entry is the identity of the respective group, and the inverse of a tuple is the tuple of inverses.

As Ferra pointed out, it isn't hard to see that in the case where $A$ is finite, the direct sum and direct product are the same, as every tuple will automatically have finite support.

However, there is another way that the notation may be construed. The internal direct sum $G=\bigoplus_{\alpha \in A}^{int}(G_\alpha)$ (the 'int' I added is not customary, I just want to stress the difference) where $G_\alpha$ is a subgroup of $G$ is meant to indicate that every element of $G$ can be uniquely expressed as a finite sum of elements from the $G_\alpha$ (finite is here to mean that the tuple of these elements has finite support; notice the similarities!).

The internal and external direct sum are closely related to each-other; it is obvious that the internal direct sum, by virtue of the uniqueness part of the expressions, means that $G$ is isomorphic to the external direct sum $\bigoplus_{\alpha\in A}{G_\alpha}$. Similarly, the external direct sum can be regarded as an internal direct sum of the groups $G_\beta\oplus \bigoplus_{\alpha\in A,\alpha\neq \beta}{\{e_\alpha\}}$, where $e_\alpha$ is the identity of the group $G_\alpha$. These isomorphisms are actually natural as well, so the distinction isn't always made explicitly, nor does it usually matter which of the two is used.


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