= Lie group
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A <Lie group> is a <group> that is also a finite-dimensional <smooth manifold>, with multiplication $G\times G\to G$ and inversion $G\to G$ both <smooth maps>. Its <tangent space> at the <identity element> carries a natural <Lie algebra> structure. Smoothness lets us differentiate the <group operation>; being a <group> means that every transformation can be composed and undone. The <dimension of a manifold> counts independent local parameters, rather than the number of elements of the <group>. See https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec02.pdf[MIT 18.745, Lecture 2, Definition 2.1 and Example 2.5].
For example, the planar <special orthogonal group> $SO(2)$ consists of <rotation matrices>
$$
R_\theta=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix},\qquad R_\alpha R_\beta=R_{\alpha+\beta},\qquad R_\theta^{-1}=R_{-\theta}.
$$
The <sine> and <cosine> entries vary smoothly. Four entries appear, but they are constrained by $R_\theta^TR_\theta=I$ and $\det R_\theta=1$: locally, one angle specifies the whole <matrix>. Multiplication adds angles, with angles differing by $2\pi$ representing the same <rotation>. Thus $SO(2)$ is an <infinite group> of <dimension of a manifold>[dimension] one, geometrically a <circle group>.
The positive-dimensional examples motivate the language of <continuous symmetry>, but the definition also admits finite <groups> with the <discrete topology> as zero-dimensional <Lie groups>. Conversely, being an <infinite group> alone says nothing about having a continuous angle parameter. The compatibility between <smooth maps> and the <group operation> is what makes differentiation useful.
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