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