By Pierre Antoine Grillet

ISBN-10: 0387715673

ISBN-13: 9780387715674

ISBN-10: 0387715681

ISBN-13: 9780387715681

A thoroughly transformed re-creation of this excellent textbook. This key paintings is geared to the wishes of the graduate scholar. It covers, with proofs, the standard significant branches of teams, jewelry, fields, and modules. Its inclusive method implies that the entire important parts are explored, whereas the extent of aspect is perfect for the meant readership. The textual content attempts to advertise the conceptual realizing of algebra as an entire, doing so with a masterful take hold of of technique. regardless of the summary subject material, the writer contains a cautious collection of very important examples, including an in depth elaboration of the extra subtle, summary theories.

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30 Chapter I. Groups In particular, FX is generated by η(X ). Proof. If a = (a1 , a2 , . . , (an ) yields a reduced word; hence a = (a1 ) . (a2 ) . · · · . (an ) = η(a1 ) . η(a2 ) . · · · . η(an ). We saw that η(x ) = η(x)−1 for all x ∈ X ; hence every a ∈ FX is a product of elements of η(X ) and inverses of elements of η(X ) . 7. Let η : X −→ FX be the canonical injection. For every mapping f of X into a group G , there is a homomorphism ϕ of FX into G unique such that f = ϕ ◦ η , namely ϕ (a1 , a2 , .

In particular, G has at most 2n elements; we do not, however, know whether these elements are distinct in G ; we might have found an equality between them if we had tried harder, or been more clever, or if, like our Lord in Heaven, we had enough time to list all consequences of the defining relations. We do, however, know that G is supposed to be isomorphic to Dn , and this provides the required alternate construction of G . 2, generated by r1 and s0 . Moreover, in Dn , the equalities r1n = s02 = 1 and s0 r1 s0 = r−1 = r1−1 hold, so that the defining relations of G hold in Dn via f : a −→ r1 , b −→ s0 .

3: Definitions. Let ϕ : A −→ B be a homomorphism of groups. The image or range of ϕ is Im ϕ = { ϕ(x) x ∈ A }. The kernel of ϕ is Ker ϕ = { x ∈ A ϕ(x) = 1 }. In the additive notation, Ker ϕ = { x ∈ A ϕ(x) = 0 } . 3, Im ϕ = ϕ(G) and Ker ϕ = ϕ −1 (1) are subgroups of B and A respectively. The kernel K = Ker ϕ has additional properties. Indeed, ϕ(x) = ϕ(y) implies ϕ(y x −1 ) = ϕ(y) ϕ(x)−1 = 1 , y x −1 ∈ K , and y ∈ K x . Conversely, y ∈ K x implies y = kx for some k ∈ K and ϕ(y) = ϕ(k) ϕ(x) = ϕ(x) .

### Abstract Algebra by Pierre Antoine Grillet

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