Group Direct Sum

The group direct sum of a sequence {G_n}_(n=0)^infty of groups G_n is the set of all sequences {g_n}_(n=0)^infty, where each g_n is an element of G_n, and g_n is equal to the identity element of G_n for all but a finite set of indices n. It is denoted

  direct sum _(n=0)^inftyG_n,

and it is a group with respect to the componentwise operation derived from the operations of the groups G_n.

This definition can easily be extended to any collection {G_i}_(i in I) of groups, where I is any finite or infinite set of indices.

If the additional condition on the identity elements is dropped, we get the definition of the group direct product. Hence, the two notions coincide whenever the set of indices is finite. Thus, for any groups G_1 and G_2,

 G_1 direct sum G_2     and     G_1×G_2

denote the same object.

If G_1 and G_2 are subgroups of the same additive group G, the equality

 G=G_1 direct sum G_2

conventionally means that every g in G has a unique decomposition g=g_1+g_2, where g_1 in G_1 and g_2 in G_2, so that G is essentially the same as the set of all ordered pairs (g_1,g_2).

See also

Direct Sum, Group Direct Product

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Group Direct Sum." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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