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The ring of fractions of an integral domain. The field of fractions of the ring of integers Z is the rational field Q, and the field of fractions of the polynomial ring ...

The field of rationals is the set of rational numbers, which form a field. This field is commonly denoted Q (doublestruck Q).

The field of reals is the set of real numbers, which form a field. This field is commonly denoted R (doublestruck R).

A group G is said to be finitely generated if there exists a finite set of group generators for G.

Let D be a planar Abelian difference set and t be any divisor of n. Then t is a numerical multiplier of D, where a multiplier is defined as an automorphism alpha of a group G ...

The fitting subgroup is the subgroup generated by all normal nilpotent subgroups of a group H, denoted F(H). In the case of a finite group, the subgroup generated will itself ...

Let a, b, and k be integers with k>=1. For j=0, 1, 2, let S_j=sum_(i=j (mod 3))(-1)^i(k; i)a^(k-i)b^i. Then 2(a^2+ab+b^2)^(2k)=(S_0-S_1)^4+(S_1-S_2)^4+(S_2-S_0)^4.

A closed embedded smooth plane curve has at least four vertices, where a vertex is defined as an extremum of curvature.

Let four lines in a plane represent four roads in general position, and let one traveler T_i be walking along each road at a constant (but not necessarily equal to any other ...

If F is a group, then the extensions G of F of order o with G/phi(G)=F, where phi(G) is the Frattini subgroup, are called Frattini extensions.