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Each Cartan matrix determines a unique semisimple complex Lie algebra via the Chevalley-Serre, sometimes called simply the "Serre relations." That is, if (A_(ij)) is a k×k ...

A theorem proved by É. Cartan in 1913 which classifies the irreducible representations of complex semisimple Lie algebras.

Let T be a maximal torus of a group G, then T intersects every conjugacy class of G, i.e., every element g in G is conjugate to a suitable element in T. The theorem is due to ...

A subring of a ring R is a subgroup of R that is closed under multiplication.

A group L is a component of H if L is a quasisimple group which is a subnormal subgroup of H.

If every component L of X/O_(p^')(X) satisfies the "Schreler property," then L_(p^')(Y)<=L_(p^')(X) for every p-local subgroup Y of X, where L_(p^') is the p-layer.

Theta(G;A)=<theta(a):a in A-1> is an A-invariant solvable p^'-subgroup of G.

The p-layer of H, L_(p^')(H) is the unique minimal normal subgroup of H which maps onto E(H/O_(p^')(H)).

For a group G, consider a subgroup H with elements h_i and an element x of G not in H, then xh_i for i=1, 2, ... constitute the left coset of the subgroup H with respect to x.

The set of elements g of a group such that g^(-1)Hg=H, is said to be the normalizer N_G(H) with respect to a subset of group elements H. If H is a subgroup of G, N_G(H) is ...