Algebra subgroup

Definition
Suppose $$G$$ is the algebra group corresponding to an associative algebra $$N$$ (that admits an algebra group) over a field $$F$$, i.e., $$G = 1 + N$$. A subgroup $$H$$ of $$G$$ is termed an algebra subgroup of $$G$$ if $$H = 1 + M$$ for some subalgebra $$M$$ of $$N$$.

Note that any algebra subgroup of an algebra group becomes an algebra group in its own right.

Facts

 * Algebra subgroup of algebra subgroup is algebra subgroup
 * Center of algebra group is algebra subgroup
 * Notion of algebra subgroup may differ for different interpretations of the same group as an algebra group