By Petre P. Teodorescu, Nicolae-A.P. Nicorovici

The suggestion of crew is key in our days, not just in arithmetic, but additionally in classical mechanics, electromagnetism, conception of relativity, quantum mechanics, concept of straightforward debris, and so on. This suggestion has constructed in the course of a century and this improvement is hooked up with the names of significant mathematicians as E. Galois, A. L. Cauchy, C. F. Gauss, W. R. Hamilton, C. Jordan, S. Lie, E. Cartan, H. Weyl, E. Wigner, and of many others. In arithmetic, as in different sciences, the straightforward and fertile rules make their method with trouble and slowly; although, this lengthy historical past could were of a minor curiosity, had the concept of team remained attached merely with fairly constrained domain names of arithmetic, these within which it happened initially. yet at this time, teams have invaded just about all mathematical disciplines, mechanics, the biggest a part of physics, of chemistry, and so on. We may perhaps say, with out exaggeration, that this is often crucial concept that happened in arithmetic because the invention of infinitesimal calculus; certainly, the concept of staff expresses, in an actual and operational shape, the imprecise and common rules of regularity and symmetry. The inspiration of crew ended in a profound figuring out of the nature of the legislation which govern typical phenomena, allowing to formulate new legislation, correcting sure insufficient formulations and supplying unitary and non contradictory formulations for the investigated phenomena.

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37) -1, In other words, we have associated the number 1 to even permutations, and -1 to odd permutations. It is easy to verify that f(o:;,O"j) = f(O"i)f(O"j). Also, the mapping f is surjective (or onto) and it follows that we have defined a surjective homomorphism (or epimorphism). Because Elements of general theory of groups 35 the mapping f is not one-to-one (or bijective). On the other hand, Ker f = His an invariant subgroup of 5:3 (note that, {I} is an invariant subgroup of {I, -I}). Thus, we may infer that the invariant subgroups of 5:3 are {aQ}, H, and 53 itself.

14) j=l where Ap, = 1 (or '£7=1 A}I1{ = J~, 6~ being Kronecker's symbol). 14) we derive the differential equations of the Lie group i = 1,2, ... , n, j = 1,2, ... , m. 15) This system of differential equations must be completely integrable, such that all the conditions have to be identically satisfied. 15) we obtain 82. n ~ ~2 = I: UaluaJ' 0 Tn 8=1 k,p=l OAk Af Aj + L. u~ Ual ~ J. -t allow us to rewrite these last equations in the form n ~ ~ ( i :::l s uUr Up OX 8 i) s 0 Up Ur OXs - i = 1,2, ...

22). 5 (ADO) Any Lie group admits a linear representation. From this theorem it follows that any Lie group is isomorphic to a group of linear transformations in a linear space, called the representation space. Therefore, each element of a Lie group can be associated to a linear operator acting in such a space. If the Lie group is of order m, then this operator depends on m real parameters aI, a2, ... 28) the identity element being e = T(O, 0, ... ,0). If we introduce a basis in the representation space, assumed to be n-dimensional, then each operator T (a 1, a2, ...