One-parameter group

In the theory of topological groups, a one-parameter subgroup is a continuous group homomorphism from the additive group of real numbers into a topological group. This is a one-parameter subgroup no subgroup in the group-theoretic sense.

Definition

One-parameter subgroups of Lie groups

Let be a Lie group, then a mapping is a one-parameter subgroup, if the picture smooth and a group homomorphism.

Examples

• The continuous group homomorphisms of the additive group of real numbers in themselves are exactly the pictures for a firm.
• The continuous group homomorphisms of the additive group of real numbers in the multiplicative group of nonzero real numbers are exactly the pictures for a firm.