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For a finite group of h elements with an n_ith dimensional ith irreducible representation, sum_(i)n_i^2=h.

A simple graph with n>=3 graph vertices in which each graph vertex has vertex degree >=n/2 has a Hamiltonian cycle.

The product of three translations along the directed sides of a triangle through twice the lengths of these sides is the identity map.

Each centered convex body of sufficiently high dimension has an "almost spherical" k-dimensional central section.

If a Sylow 2-subgroup T of G lies in a unique maximal 2-local P of G, then P is a "strongly embedded" subgroup of G, and G is known.

When all the points P on one line are related by an isometry to all points P^' on another, the midpoints of the segments PP^' are either distinct and collinear or coincident.

If a prime number divides a norm but not the bases of the norm, it is itself a norm.

A finite division algebra is a field.

The geometric centroid of a quadrilateral lamina is the center of its Wittenbauer's parallelogram.

Let x=(x_1,x_2,...,x_n) and y=(y_1,y_2,...,y_n) be nonincreasing sequences of real numbers. Then x majorizes y if, for each k=1, 2, ..., n, sum_(i=1)^kx_i>=sum_(i=1)^ky_i, ...