# One-parameter subgroup

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of a Lie group $G$ over a normed field $K$

An analytic homomorphism of the additive group of the field $K$ into $G$, that is, an analytic mapping $\alpha : K \rightarrow G$ such that

$$\alpha ( s + t) = \alpha ( s) \alpha ( t),\ s, t \in K.$$

The image of this homomorphism, which is a subgroup of $G$, is also called a one-parameter subgroup. If $K = \mathbf R$, then the continuity of the homomorphism $\alpha : K \rightarrow G$ implies that it is analytic. If $K = \mathbf R$ or $\mathbf C$, then for any tangent vector $X \in T _ {e} G$ to $G$ at the point $e$ there exists a unique one-parameter subgroup $\alpha : K \rightarrow G$ having $X$ as its tangent vector at the point $t = 0$. Here $\alpha ( t) = \mathop{\rm exp} tX$, $t \in K$, where $\mathop{\rm exp} : T _ {e} G \rightarrow G$ is the exponential mapping. In particular, any one-parameter subgroup of the general linear group $G = \mathop{\rm GL} ( n, K)$ has the form

$$\alpha ( t) = \mathop{\rm exp} tX = \ \sum _ {n = 0 } ^ \infty { \frac{1}{n! } } t ^ {n} X ^ {n} .$$

If $G$ is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of $G$ are the geodesics passing through the identity $e$.

#### References

 [1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) [2] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) [3] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)