Intuitive definition: real group of rotations + reflections.

Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.

We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".

By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".

Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

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If illegal porn were to ever be found, we would be unable to acknowledge that, and would just have to silently remove it. Of course, the reproducible nature of Bitcoin Inscription Indexer means that anyone who regenerates the data would immediately see such entries in the diff.

Another issue is that it can be quite hard to determine if porn is legal or illegal, e.g. it can be hard to distinguish legal an illegal ages or revenge vs consensual porn, especially at the low resolutions that you may expect to find embedded in the blockchain. We are generally going to be quite strict about this, and in case of uncertainty on porn legality we will censor first.

OK the list:

From Section "Lie algebra of a isometry group" we reach:

Group of rotations of a rigid body.

Like orthogonal group but without reflections. So it is a "special case" of the orthogonal group.

This is a subgroup of both the orthogonal group and the special linear group.

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Based on the $L_x$,$L_y$ and $L_z$ derived at Lie algebra of $SO(3)$ we can calculate the Lie bracket as:

Has $SU(2)$ as a double cover.

Group of the unitary matrices.

Complex analogue of the orthogonal group.

One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. $U(1)$ is the unit circle.

Diffeomorphic to the 3 sphere.

The $U(1)$ unitary group is one very over-generalized way of looking at it :-)

Analysis of some of them follows.

The complex analogue of the special orthogonal group, i.e. the subgroup of the unitary group with determinant equals exactly 1 instead of an arbitrary complex number with absolute value equal 1 as is the case for the unitary group.