Homework 26

Read: Read Section 7.2.

Please try (but do not turn in): 7.7, 7.9, 7.17, 7.19, 7.22, 7.24, 7.25

• For 7.7, if $|g| = m$, then you know that $g^m = e$ (and that $m$ is the smallest positive integer with this property); if you’re stuck, try applying $\phi$ to both sides. Corollary 4.25 should help.

• For 7.22, 7.24, 7.25, you are trying to show $G_1/N\cong G_2$ for different choices of $G_1$, $N$, and $G_2$. This can be done using the First Isomorphism Theorem by following these steps: (1) define a homomorphism $\phi:G_1 \rightarrow G_2$ (this can be the hardest step…unless it’s already done for you), (2) show that $\phi(G_1) = G_2$, and (3) show that $\ker(\phi) = N$. Once you’ve done that, the First Isomorphism Theorem tells you that $G_1/N\cong G_2$. (Note that Problems 7.3 and 7.9 are helpful for 7.25.)