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A group acts freely if there are no group fixed points. A point which is fixed by every group element would not be free to move.

Let F be a field of field characteristic p. Then the Frobenius automorphism on F is the map phi:F->F which maps alpha to alpha^p for each element alpha of F.

The center of a group is the set of elements which commute with every element of the group. It is equal to the intersection of the centralizers of the group elements.

A ring that is commutative under multiplication, has a multiplicative identity element, and has no divisors of 0. The integers form an integral domain.

The number of elements greater than i to the left of i in a permutation gives the ith element of the inversion vector (Skiena 1990, p. 27).

For a group G, consider a subgroup H with elements h_i and an element x of G not in H, then xh_i for i=1, 2, ... constitute the left coset of the subgroup H with respect to x.

Let H be a subgroup of G. A subset T of elements of G is called a left transversal of H if T contains exactly one element of each left coset of H.

A mathematical object which consists of a set of a single element, making it a 1-tuple. The yin-yang is also known as the monad.

An algebraic extension K over a field F is a purely inseparable extension if the algebraic number minimal polynomial of any element has only one root, possibly with ...

An action which adds a single element to the top of a stack, turning the stack (a_1, a_2, ..., a_n) into (a_0, a_1, a_2, ..., a_n).