Abstract Algebra by Pierre Antoine Grillet PDF

By Pierre Antoine Grillet

ISBN-10: 0387715673

ISBN-13: 9780387715674

ISBN-10: 0387715681

ISBN-13: 9780387715681

A thoroughly transformed new version of this significant textbook. This key paintings is geared to the desires of the graduate scholar. It covers, with proofs, the standard significant branches of teams, earrings, fields, and modules. Its inclusive process signifies that all the precious components 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 seize of technique. regardless of the summary subject material, the writer features a cautious collection of vital examples, including an in depth elaboration of the extra refined, summary theories.

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Sample text

The usual practice is to identify the elements of A or B and the corresponding one-letter words; then A and B are subgroups of A B , and the latter is generated by A ∪ B . , x y rather than x . y . Various other symbols are used instead . of Universal property. 6, the free product of A and B is also the free product of Im ι and Im κ (up to isomorphism). If A ∩ B =/ 1, then the free product of A and B is defined (up to isomorphism) as the free product of any A ∼ = A and B ∼ = B such that A ∩ B = 1, with injections A −→ A −→ A B and B −→ B −→ A B .

Show that FX ∼ = Z if X has just one element. 4. 7 characterizes the free group on X up to isomorphism. (Let F be a group and let j : X −→ F be a mapping. Assume that for every mapping f of X into a group G , there is a homomorphism ϕ of F into G unique such that f = ϕ ◦ j . ) 5. Show that every mapping f : X −→ Y induces a homomorphism Ff : FX −→ FY unique such that Ff ◦ η X = ηY ◦ f (where η X , ηY are the canonical injections). Moreover, if f is the identity on X , then Ff is the identity on FX ; if g ◦ f is defined, then Fg◦ f = Fg ◦ Ff .

1. If ϕ : A −→ B and ψ : B −→ C are homomorphisms of groups, then so is ψ ◦ ϕ : A −→ C . Moreover, the identity mapping 1G on a group G is a homomorphism. Homomorphisms preserve identity elements, inverses, and powers, as readers will gladly verify. In particular, homomorphisms of groups preserve the constant and unary operation as well as the binary operation. 2. If ϕ : A −→ B is a homomorphism of groups (written multiplicatively), then ϕ(1) = 1, ϕ(x −1 ) = ϕ(x) −1 , and ϕ(x n ) = ϕ(x) n , for all x ∈ A and n ∈ Z .

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Abstract Algebra by Pierre Antoine Grillet


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