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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.

A group given by G/phi(G), where phi(G) is the Frattini subgroup of a given group G.

The intersection phi(G) of all maximal subgroups of a given group G.

The free product G*H of groups G and H is the set of elements of the form g_1h_1g_2h_2...g_rh_r, where g_i in G and h_i in H, with g_1 and h_r possibly equal to e, the ...

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.

A finite extension K=Q(z)(w) of the field Q(z) of rational functions in the indeterminate z, i.e., w is a root of a polynomial a_0+a_1alpha+a_2alpha^2+...+a_nalpha^n, where ...