Algebra I: Chapters 1-3 by Nicolas Bourbaki

By Nicolas Bourbaki

This softcover reprint of the 1974 English translation of the 1st 3 chapters of Bourbaki’s Algebre offers an intensive exposition of the basics of basic, linear, and multilinear algebra. the 1st bankruptcy introduces the elemental items, comparable to teams and jewelry. the second one bankruptcy experiences the homes of modules and linear maps, and the 3rd bankruptcy discusses algebras, specially tensor algebras.

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H . THEOREM 1. Let G be a group. (a) Every homogeneous G-set is isomorphic to a homogeneous G-set of the form G/H H is a subgroup of G. (b) Let H and H' be two subgroups of G. The G-sets G/H and G/H' are isomorphic if and only if H and H' are conjugate. Where As a homogeneous G-set is non-empty, assertion (a) follows from Proposition 6. We show (b). L e t 3 G / H -+ G/H' be a G-set isomorphism. The subgroup H is the stabilizer of H and hence, by transport of structure, the stabilizer of an 58 59 Condition (ii) is also equivalent to the following condition: element of G/H'.

In any group G, if the (monogenous) subgroup generated by an element E G is of finite order d, x is called an element of order d ; the number d is therefore the least integer > 0 such that xd = e ; if the subgroup generated by x is infinite, xis said to be of infinite order. These definitions,together with Proposition 4 x 51 I MONOID OPERATING ON A SET ALGEBRAIC STRUCTURES (no. 4), imply in particular that in a finite group G the order of every element of G is a divisor of the order of G. PROPOSITION 21.

DEFINITION 14. POUPSof G. G is called the internal restricted sum (or restricted sum) o f thefamily ofsubgroups (HI) ifevery element of H, is permutable with every element ofH, for j # i and the unique homomorphism o f a Hi into G whose restriction to each o f the H, is the canonical injection is an isomorphism. When I is finite, we also say, by an abuse of language, internal direct product (Or direct product, or product) instead of internal restricted sum. 10 ALGEBRAIC STRUCTURES MONOGENOUS GROUPS subgroup H of G for which there exists a stable subgroup H’ of G such that G is the direct product of H and H‘ is called a directfactor of G.

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