Math 360, Fall 2021, Assignment 15

Carefully state the following theorems (you do not need to prove them):

1. Corollary (of Lagrange's Theorem) concerning groups of prime order.
2. Corollary (of Lagrange's Theorem) concerning $g^{\left\lvert G\right\rvert}$.

Solve the following problems:

1. Describe all subgroups of $S_3$, and draw the subgroup diagram. (Hint: apart from the trivial and improper subgroups, the only possible orders are $2$ and $3$. Both of these are prime numbers.)
2. Describe all subgroups of $D_5$, and draw the subgroup diagram. (Use the same technique as in the previous problem.)
3. Show that, by contrast, $D_4$ has non-trivial proper subgroups which are not cyclic; hence, the technique of the previous two problems would have missed some subgroups.

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Theorems:

1. Corollary (of Lagrange's Theorem) concerning groups of prime order: Suppose $|G| = p$ for prime $p$. Then $G$ is isomorphic to $(\mathbb Z_{p}, +)$.
2. Corollary (of Lagrange's Theorem) concerning $g^{\left\lvert G\right\rvert}$: Let $G$ be any finite group, and let $g \in G$. Then $g^{|G|} = e$

Problems:

3. Imagine a square, and let $\mu_1$ and $\mu_2$ be the reflections in the lines through the midpoints of pairs of opposite edges. Then $\mu_1$ and $\mu_2$ together generate a subgroup with four elements, isomorphic to the Klein 4-group, which is not cyclic.