There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as $GF(n)$.

Every element of a finite field satisfies $x^{order}=x$.

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

Discovery: induced pluripotent stem cell.

Elliptic curve point addition is the group operation of an elliptic curve group, i.e. it is a function that takes two points of an elliptic curve as input, and returns a third point of the elliptic curve as its output, while obeying the group axioms.

The operation is defined e.g. at en.wikipedia.org/w/index.php?title=Elliptic_curve_point_multiplication&oldid=1168754060#Point_operations. For example, consider the most common case for two different points different. If the two points are given in coordinates:then the addition is defined in the general case as:with some slightly different definitions for point doubling $P+P$ and the identity point.

This definition relies only on operations that we know how to do on arbitrary fields:and it therefore works for elliptic curves defined over any field.

Just remember that:means:and that $y^{-1}$ always exists because it is the inverse element, which is guaranteed to exist for multiplication due to the group axioms it obeys.

The group function is usually called elliptic curve point addition, and repeated addition as done for DHKE is called elliptic curve point multiplication.

web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:TODO why this non standard formulation?

This construction takes as input:and it produces an elliptic curve over a finite field of order $p$ as output.

The constructions is used in the Birch and Swinnerton-Dyer conjecture.

To do it, we just convert the coefficients $a$ and $b$ from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.

For example, suppose we have $a=3/4$ and we are using $p=11$.

For the denominator $4$, we just use the multiplicative inverse, e.g. supposing we havewhere $4^{-1}=3\mathrm{m}\mathrm{o}\mathrm{d}11$ because $4\times 3=1\mathrm{m}\mathrm{o}\mathrm{d}11$, related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p

The Equation "Definition of the elliptic curves" and definitions on elliptic curve point addition both hold directly.

Suppose we have a given permutation group that acts on a set of n elements.

If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.

Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group

TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.

Bibliography:

Directly modelled by group.

For continuous symmetries, see: Lie group.