Download e-book for kindle: Algebra (Prindle, Weber and Schmidt Series in Advanced by Mark Steinberger

By Mark Steinberger

ISBN-10: 0534936784

ISBN-13: 9780534936785

The rationale of this e-book is to introduce readers to algebra from some extent of view that stresses examples and type. every time attainable, the most theorems are handled as instruments that could be used to build and learn particular kinds of teams, earrings, fields, modules, and so forth. pattern structures and classifications are given in either textual content and routines.

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List the homomorphisms from Z5 to Z6 . 5. Show that the order of m in Zn is n/(m, n). Deduce that the order of each element divides the order of Zn . Deduce that every non-identity element of Zp has order p, for any prime p. † 6. Let G be a group and let x ∈ G. Show that o(x−1 ) = o(x). ) CHAPTER 2. GROUPS: BASIC DEFINITIONS AND EXAMPLES 34 7. Show that a group G is cyclic if and only if there is a surjective homomorphism f : Z → G. † 8. Let f : G → G be a homomorphism. (a) Let H ⊂ G be a subgroup.

Define f (H) ⊂ G by f (H) = {f (x) | x ∈ H}. Show that f (H) is a subgroup of G . If ker f ⊂ H, show that f −1 (f (H)) = H. (c) Suppose that H ⊂ im f . Show that f (f −1 (H )) = H . Deduce that there is a one-to-one correspondence between the subgroups of im f and those subgroups of G that contain ker f . What does this tell us when f is onto? 9. Let f : G → G be a homomorphism and let x ∈ G. Show that f ( x ) = f (x) . 10. Show that every subgroup of Zn is cyclic. 11. Show that Zn has exactly one subgroup of order d for each d dividing n, and has no other subgroups.

Prkk and q1s1 . . qlsl of m as above. We argue by induction on t = r1 + · · · + rk . If t = 1, then m = p1 , a prime. But then p1 is divisible by q1 , and since both are prime, they must be equal. We then have p1 (1 − q1s1 −1 . . qlsl ) = 0, so that (1 − q1s1 −1 . . qlsl ) = 0, and hence q1s1 −1 . . qlsl = 1. Since the qi are prime and since no positive number divides 1 other than 1 itself, we must have l = 1 and s1 = 1. Suppose t > 1. 16, together with an induction on s1 + · · · + sl , shows that p1 must divide qi for some i.

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Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) by Mark Steinberger

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