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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]].\nA *one-parameter subgroup* of $G$ is a continuous [[group homomorphism|0JYTG5K9Y4]] $\\gamma \\colon \\R \\to G$ from the additive group of real numbers.\nSo $\\gamma$ is continuous and\n$$\\gamma(s + t) = \\gamma(s)\\gamma(t) \\quad \\text{for all } s, t \\in \\R.$$\n\n## Remarks\n\nDespite the name, a one-parameter subgroup is the map $\\gamma$, not its image.\n\nExample: for a real $n \\times n$ matrix $A$, the map $\\gamma(t) = e^{tA}$ into $GL_n(\\R)$, given by the [[matrix exponential|020M04FN1H]], is continuous, and $e^{(s + t)A} = e^{sA} e^{tA}$ because $sA$ and $tA$ commute.\n\nNon-example: $\\gamma(t) = \\begin{pmatrix} 1 \u0026 t \\\\ 0 \u0026 1 + t^2 \\end{pmatrix}$ is a continuous curve in $GL_2(\\R)$ with $\\gamma(0) = I$, but it is not a homomorphism.\nThe $(2,2)$ entry of $\\gamma(1)\\gamma(1)$ is $4$, while that of $\\gamma(2)$ is $5$.\n\nIf $G$ is a [[Lie group|7JBRGCVJRB]], every one-parameter subgroup is smooth.\nThen $\\gamma \\mapsto \\gamma'(0)$ is a bijection from the one-parameter subgroups of $G$ onto the tangent space $T_eG$.\n",
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