A fundamental idea in the contemporary study of geometry, especially applied differential geometry, is that of the Lie group, pronounced "lee" group. The theory of Lie groups formalizes a very broad class of related geometric ideas about symmetry: rotation, translation, and perspective.
Before we discuss Lie groups, however, we must first address two related concepts: that of the group and that of the manifold.
A group is defined as a set \( G \) with a binary operator \( \circ : G \times G \rightarrow G \). For \( G \) to be a group, the ordered pair \( ( G, \circ ) \) must obey three properties: closure, identity, and inverse properties.
The closure property is perhaps blatantly obvious, but it is appropriate to state it formally. For \( G \) to be a group under \( \circ \), it must be true that, for all \( g_1, g_2 \in G \), we have \( g_1 \circ g_2 \in G \). In other words, if \( \circ \) is defined in such a way that its outputs are not in \( G \), then \( G \) is not a group.
For \( G \) to be a group under \( \circ \), there must exist an identity element \( e \in G \) such that \( e \circ g = g \circ e = g \) for all \( g \in G \).
The final property required is the inverse property. To be a group under \( \circ \), \( G \) must contain for every \( g \) an inverse element \( g^{-1} \) such that \( g \circ g^{-1} = g^{-1} \circ g = e \).
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