# One-parameter group not a group? Why?

So one-parameter group $G$ is defined with a continuous group homomorphism $\phi: \mathbb{R} \rightarrow G$.

But according to the texts I read, they say that $G$ must be distinguished from groups as $G$ is not a group. So my wonder is, but there is group homomorphism there... So what is going on?

• @anon I think you might as well have posted that as an answer. – Hagen von Eitzen Apr 7 '13 at 11:26

• For example, $\phi\colon\mathbb R\to \mathbb C^\times$, $t\mapsto e^{it}$ describes (has as image) the subgroup $S^1$, but so does $t\mapsto e^{i\omega t}$ for any nonzero real $\omega$. For the image $S^1$, $\omega$ does not play a role, but we do have a different one-parameter subgorup for each possible coice of $\omega$. – Hagen von Eitzen Apr 7 '13 at 12:03