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