Group Direct Product


Given two groups G and H, there are several ways to form a new group. The simplest is the direct product, denoted G×H. As a set, the group direct product is the Cartesian product of ordered pairs (g,h), and the group operation is componentwise, so


For example, R×R is isomorphic to R^2 under vector addition. In a similar fashion, one can take the direct product of any number of groups by taking the Cartesian product and operating componentwise.

Note that G is isomorphic to the subgroup of elements (g,e_H) where e_H is the identity element in H. Similarly, H can be realized as a subgroup. The intersection of these two subgroups is the identity (e_G,e_H), and the two subgroups are normal.


Like the ring direct product, the group direct product has the universal property that if any group X has a homomorphism to G and a homomorphism to H, then these homomorphisms factor through G×H in a unique way.

If one has group representations R_G of G and R_H of H, then there is a representation R_G tensor R_H sometimes called the external tensor product, given by the tensor product  tensor . In this case, the group character satisfies

 chi(g tensor h)=chi_(R_G)(g)chi_(R_H)(h).

See also

Cartesian Product, External Tensor Product, Group Representation, Homomorphism, Subgroup, Universal Property

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Group Direct Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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