Math 71 Homework Assignment 20 October 1999

• p. 90: 41

• p. 102: 3 [Hint: Use the Second Isomorphism Theorem], and do not assume any of the groups are finite.

• New Proof of Second Isomorphism Theorem (Theorem 18, p. 98): Let G be a group with A, B subgroups of G and with B ⊴ G. Then A ∩ B ⊴ A and AB/B ≅ A/A ∩ B.

Be sure to verify that AB is a subgroup of G, B ⊴ AB, and A ∩ B ⊴ A.

Then proceed with a proof by justifying that φ : A → AB/B induced by a natural composition of maps A → AB → AB/B (a ↦ a · 1 ↦ aB) is a surjective homomorphism. Computing the kernel of φ and applying the first isomorphism theorem should yield the result.