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 $B \trianglelefteq G$. Then $A \cap B \trianglelefteq A$ and $AB/B \cong A/A\cap B$.

  Be sure to verify that $AB$ is a subgroup of $G$, $B \trianglelefteq AB$, and $A \cap B \trianglelefteq A$.

  Then proceed with a proof by justifying that $\varphi: A \to AB/B$ induced by a natural composition of maps $A \to AB \to AB/B$ ( $a \mapsto a\cdot 1 \mapsto aB$) is a surjective homomorphism. Computing the kernel of $\varphi$ and applying the first isomorphism theorem should yield the result.