MODERN ALGEBRA (MATH 210) PROBLEM SET 5

The proof of the Zassenhaus Lemma (p. 279), and hence the Jordan-Hölder Theorem, relies on the certain facts unproved in the text. Rotman defines a map of sets

ϕ : A(A^{∗} ∩ B^{∗}) → \frac{A^{∗} ∩ B^{∗}}{(A ∩ B^{∗})(A^{∗} ∩ B)}

by ax ↦ xD, where a ∈ A and x ∈ A^{∗} ∩ B^{∗}. Show that (i) ϕ is a group homomorphism, (ii) ϕ is surjective, and (iii) Ker ϕ = A(A^{∗} ∩ B).

Do nine of the following ten problems from the text: 5.18 , 5.33 , 5.34 , 5.37 , 5.39 , 5.40 , 5.41 , 5.42 , 5.45 , 5.49 .

The set is due Tuesday, November 19 at 3:30 pm in Pierre Albin’s mailbox.

Date: Tuesday, November 12, 2002.