By Arthur Cayley

This quantity is made out of electronic pictures from the Cornell college Library historic arithmetic Monographs assortment.

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K . If we place a lexicographic linear order ≤ on the set of functionals α, the CartanWeyl Theorem asserts that there exists an irreducible representation ρ of G having α as its highest weight (with respect to the order ≤) if ] and only if 2[α,σ [σ,σ ] is an integer for every σ ∈ , and w(α) ≤ α for every permutation w in the Weyl group of G relative to H. Cartan’s criterion of semisimplicity A Lie algebra L is semisimple if and only if the Killing form K of L is nondegenerate. Carter subgroup Any finite solvable group contains a self-normalizing, nilpotent subgroup, called a Carter subgroup.

Closed subsystem Let M be a character module. Let s be a subset of a root system r, and let s be the submodule of M generated by s. If s ∩ r = s, then s is called a closed subsystem. See character module. closed under operation A set S, with a binary operation ∗ such that a ∗ b ∈ S for all a, b ∈ S. closure property The property of a set of being closed under a binary operation. See closed under operation. coalgebra Let ρ and ρ be linear mappings defined as ρ : V → V ⊗F V , and ρ : V → F , where F is a field, V is vector space over F , and ⊗ denotes tensor product.

Am Also called column matrix. combination of things When r objects are selected from a collection of n objects, the r selected objects are called a combination of r objects from the collection of n objects, provided that the selected objects are all regarded as having equal status and not as being in any particular order. ) The number of different combinations of r obn jects from n is denoted by n Cr or and r n! equals . (n − r)! commensurable Two non-zero numbers a and b such that a = nb,√ for some rational num√ ber n.