Read: Continue with 4.1.
Turn in: 4.38, 4.42, 4.43
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It should be emphasized that 4.41 is a very important result about cyclic groups. It’s definitely worth memorizing, even though you’re not being asked to prove it.
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Your proof of 4.43 might be quite short; you want to make use of Theorem 4.41 (and the definition of \(n\mathbb{Z}\)).
Extra practice: 4.39, 4.40, 4.41
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On 4.40, you just need to properly link together Theorems 4.27, 4.29, and 4.39. Also, remember that Theorem 3.56 implies that if $G_1 \cong G_2$ and $G_2 \cong G_3$, then $G_1 \cong G_3$.
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4.41 is quite challenging. Assuming $G$ is cyclic and $H$ is a subgroup, you aim to find a generator for $H$. Since $G$ is cyclic, it has some generator $a$. This means that the elements of $G$, hence the elements of $H$, can be written as powers of $a$, but which one might be a generator for $H$? If $H$ is the trivial subgroup, then $H$ is generated by $e$, and you are done. If $H$ is nontrivial, then one idea is to consider $a^k$ where $k$ is the smallest positive integer such that $a^k\in H$.