Cross product

Summary

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are not linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the

Official source

https://en.wikipedia.org/wiki/Cross_product

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The minimal faithful permutation degree (G) of a finite group G is the least nonnegative integer n such that G embeds in the symmetric group Sym(n). Clearly (G x H) (G) + (H) for all finite groups G and H. In 1975, Wright ([10]) proved that equality occurs when G and H are nilpotent and exhibited an example of strict inequality where G x H embeds in Sym(15). In 2010 Saunders ([7]) produced an infinite family of examples of permutation groups G and H where (G x H) < (G) + (H), including the example of Wright's as a special case. The smallest groups in Saunders' class embed in Sym(10). In this article, we prove that 10 is minimal in the sense that (G x H)=(G) + (H) for all groups G and H such that (G x H)

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