One-parameter group

Definition

Let $G$ be a topological group. A one-parameter group or one-parameter subgroup; in $G$ is a continuous homomorphism of groups from the additive group of real numbers to $G$, i.e., a continuous map:

$\phi :\mathbb{R}\to G$

such that for all $s,t\in \mathbb{R}$$\phi (s+t)$ is the product (in $G$) $\phi (s)\phi (t)$.

We can verify from this that $\phi (-s)=\phi (s{)}^{-1}$ for all $s\in \mathbb{R}$ and that $\phi (0)$ is the identity element of $G$.

Note that the image of $\phi$ is an abelian subgroup of $G$. However, the term one-parameter group or one-parameter subgroup is used for the mapping $\phi$ itself, not for the image of the mapping.

Typically, trivial homomorphisms are excluded from the definition of one-parameter group.

Caveats

1. Different one-parameter groups may have the same image, but differ in terms of the parametrizations. For instance, one of the one-parameter groups may be obtained composing a scalar multiplication with the other. In this case, the groups have the same image but are not the same as one-parameter groups.
2. The homomorphism need not be injective. In general, assuming the image of the mapping is a T0 topological group, the mapping is either injective or trivial or its kernel is the set of integer multiples of some number. In the last case, the quotient group is (as an abstract group) isomorphic to the circle group.
3. The quotient topology on the one-parameter group from the topology of $\mathbb{R}$ may be strictly finer than the subspace topology from $G$. This happens, for instance, for a line of irrational slope on a torus.