Inequalities between the Two Kinds of Eigenvalues of a Linear Transformation.

an extension (T, y6) of Q such that T contains E as local subgroup and {1 E = q| E. We call the pair (Y, X) elementary with respect to Q if every extension of X is extensible over Q from Y. We can now state a group-theoretic equivalent of the condition P2( Y, X) = 0. THEOREM 3. Let X, Y be local subgroups of a group Q such that (1) Y is a local subgroup of X; (2) Q is simply connected relative to Y; (3) X contains no elements of order 2. A necessary and sufficient condition thtat (Y, X) be elementary with respect to Q is that P2(Y, X) = 0 With the aid of Theorem 3, it is possible to give a proof of the existence of Lie groups in the large based on the vanishing of the second homotopy group rather than on the theorem of E. Levi.3 The details will appear elsewhere.