Diagonal subgroup

The subgroup of a Cartesian power of a given group $ G $ consisting of all elements with identical components. For instance, the diagonal group of the product $ G \times G $ is the group of pairs $ ( g , g ) $, $ g \in G $.

Comments

The phrase diagonal group (or diagonal subgroup) is also used for the subgroup scheme $ D ( n, k) $ of $ \mathop{\rm GL} ( n, k) $ over a field $ k $ whose points with values in a $ k $- algebra $ R $ are the diagonal invertible matrices with coefficients in $ R $.

Let $ \Gamma $ be a commutative group. The functor $ D ( \Gamma ) : R \rightarrow {\mathcal G} {\mathcal r} ( \Gamma , R ^ {*} ) $ from commutative rings with unit element to groups then defines a group scheme. Here $ R ^ {*} $ is the group of invertible elements of $ R $ and $ {\mathcal G} {\mathcal r} $ is the category of groups. Group schemes isomorphic to such group schemes are called diagonizable group schemes.

References

[a1]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1970)