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For a subgroup H of a group G, the index of H, denoted (G:H), is the cardinal number of the set of left cosets of H in G (which is equal to the cardinal number of the set of ...

The symmetric group S_n of degree n is the group of all permutations on n symbols. S_n is therefore a permutation group of order n! and contains as subgroups every group of ...

Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of (p_i,q_i) planes. It is the generalization of an area-preserving map. ...

The third Morley cubic is the triangle cubic with trilinear equation It passes through Kimberling centers X_n for n=1, 357, 358, 1136, and 1137.

A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus. ...

A totative is a positive integer less than or equal to a number n which is also relatively prime to n, where 1 is counted as being relatively prime to all numbers. The number ...

Every real number is negative, 0, or positive. The law is sometimes stated as "For arbitrary real numbers a and b, exactly one of the relations a<b, a=b, a>b holds" (Apostol ...

The plane curve given by the equation axy = (a+x)(a-x)(2a-x) (1) = x^3-2ax^2-a^2x+2a^3, (2) which, solving for y, gives y=((a+x)(a-x)(2a-x))/(ax). (3) The plots above are for ...

The tridiminished icosahedron is Johnson solid J_(63). Interestingly, its skeleton has the same degree sequence as the triangular cupola.

cos(pi/(12)) = 1/4(sqrt(6)+sqrt(2)) (1) cos((5pi)/(12)) = 1/4(sqrt(6)-sqrt(2)) (2) cot(pi/(12)) = 2+sqrt(3) (3) cot((5pi)/(12)) = 2-sqrt(3) (4) csc(pi/(12)) = sqrt(6)+sqrt(2) ...