Lie Group

We start with a group, G.

• We determine that G is a differentiable manifold.
• We have Multiplication (if we multiply an element in G by a scalar, the result is also an element in G)
• We have Inverses.

This is enough to declare that G is a continuous group and we are almost there.

If multiplication and the taking of inverses are both smooth, then G is a lie group.

Appendix A

How can inverses be smooth?

We can create a function that maps element x to the element that is its inverse, 1/x, and if the derivative function is continuous then the taking of inverses is smooth.

Note: in the above we write 1/x as a symbol for an element.