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Diagonal subgroup

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Diagonal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_subgroup&oldid=46643
This article was adapted from an original article by Yu.I. Merzlyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article