View source for Diagonal subgroup

The subgroup of a Cartesian power of a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315401.png" /> consisting of all elements with identical components. For instance, the diagonal group of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315402.png" /> is the group of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315404.png" />.



====Comments====
The phrase diagonal group (or diagonal subgroup) is also used for the subgroup scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315405.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315406.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315407.png" /> whose points with values in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315408.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d0315409.png" /> are the diagonal invertible matrices with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154010.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154011.png" /> be a commutative group. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154012.png" /> from commutative rings with unit element to groups then defines a [[Group scheme|group scheme]]. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154013.png" /> is the group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031540/d03154015.png" /> is the category of groups. Group schemes isomorphic to such group schemes are called diagonizable group schemes.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Demazure,   P. Gabriel,   "Groupes algébriques" , '''1''' , North-Holland  (1970)</TD></TR></table>