By Mark Steinberger
The purpose of this publication is to introduce readers to algebra from some extent of view that stresses examples and class. every time attainable, the most theorems are taken care of as instruments which may be used to build and learn particular kinds of teams, earrings, fields, modules, and so on. pattern buildings and classifications are given in either textual content and workouts.
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Additional resources for Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics)
Let m and n be relatively prime. Show that Zmn is isomorphic to Zm × Zn . 5. Let n = pr11 . . prkk , where p1 , . . , pk are distinct primes. Show that Zn ∼ = Zpr11 × · · · × Zprkk . 6. Show that if m and n are not relatively prime, then Zm × Zn is not cyclic. 7. Show that D4 is isomorphic to Z2 × Z2 . 8. Show that D12 is isomorphic to D6 × Z2 . 9. Show that if k is odd, then D4k is isomorphic to D2k × Z2 . 10. Show that G × H ∼ = H × G. 11. Show that (G × H) × K ∼ = G × (H × K) ∼ = G × H × K.
Then every element of the image of f ◦ ι1 commutes with every element of the image of f ◦ ι2 . Moreover, f is uniquely determined by its restrictions to G × e and e × H. Conversely, given homomorphisms g1 : G → K and g2 : H → K such that each element in the image of g1 commutes with each element in the image of g2 , then there is a unique homomorphism f : G × H → K with f ◦ ιi = gi for i = 1, 2. Proof Since (g, h) = (g, e) · (e, h), f (g, h) = f ◦ ι1 (g) · f ◦ ι2 (h). Thus, f is determined by the f ◦ ιi , which are determined by the restriction of f to the stated subgroups.
1. (The Euclidean Algorithm2 ) Let m and n be integers, with n > 0. Then there are integers q and r, with 0 ≤ r < n, such that m = qn + r. Proof First, we assume that m ≥ 0, and argue by induction on m. If m < n, we may take q = 0 and r = m. If m = n, we take q = 1 and r = 0. Thus, we may assume that m > n and that the result holds for all non-negative integers less than m. In particular, the induction hypothesis gives m−1=q n+r , for integers q and r with 0 ≤ r < n. If r < n − 1, we may take q = q and r = r + 1 for the desired result.
Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics) by Mark Steinberger