A finite group G is generated through the pseudo form of this operation if and only if the ring of invariants of G acting on a finite-dimensional vector space form a polynomial ring.
A finite group G is generated through the pseudo form of this operation if and only if the ring of invariants of G acting on a finite-dimensional vector space form a polynomial ring. For an n-dimensional symmetric bilinear space over a field with characteristic not equal to two, every element of its orthogonal group is a composition of at most n of these operations. For a nonzero vector v in a vector space, a transformation describing this operation applied to x is equal to x minus quantity two times inner product of x and v divided by inner product of v with itself, all times v. That transformation is the (*) Householder transformation. For the dihedral group D sub n, if n is even, the set of these fall into two conjugacy classes. Matrices describing this operation are orthogonal and have determinant equal to negative one. The composition of an even number of instances of this operation yields a rotation. For 10 points, name this operation which creates mirror images.
A finite group G is generated through the pseudo form of this operation if and only if the ring of invariants of G acting on a finite-dimensional vector space form a polynomial ring.
reflection [the first theorem is Chevalley-Shephard-Todd; the second is Cartan-DieudonnÃ©]