Dimension of extension group is sum of dimensions of normal subgroup and quotient group
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Contents
Statement
Statement for algebraic groups
Suppose is an algebraic group,
is a closed normal subgroup and
is the quotient group. Both
and
acquire algebraic group structures from the algebraic group structure of
. We then have the following relationship between the dimensions of
:
In particular, is a finite-dimensional algebraic group if and only if both
and
are.
Statement for Lie groups
Suppose is a Lie group,
is a closed normal subgroup and
is the quotient group. Both
and
acquire Lie group structures from the Lie group structure of
. We then have the following relationship between the dimensions of
: