Math 360, Fall 2019, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Read:

1. Section 8.

Carefully define the following terms, then give one example and one non-example of each:

1. Permutation (of a set $S$).
2. $\mathrm{Sym}(S)$ (the symmetric group on $S$).
3. $S_n$ (the symmetric group on $n$ letters).
4. Two-row notation (for the permutation $\pi$).
5. Order (of a group).
6. Permutation model (for a group $G$).
7. Finite permutation model (for a group $G$).
8. $D_n$ (the $n$th dihedral group).

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

1. Theorem concerning the order of $S_n$.
2. Cayley's Theorem.

Solve the following problems:

1. Section 8, problems 1, 3, 5, 7, 11, 13, and 17.
2. Suppose $G$ and $H$ are groups, and that $\phi:G\rightarrow H$ is a function satisfying $\phi(g_1g_2)=\phi(g_1)\phi(g_2)$. Prove that the image of $\phi$ is a subgroup of $H$. (Hint: Let $e_G$ and $e_H$ denote that identity elements of $G$ and $H$, respectively. Begin by showing that $\phi(e_G)=e_H$, and that $\phi(g^{-1})=(\phi(g))^{-1}$.)
3. (Challenge) In a previous assignment, you investigated the symmetry group of the methane molecule. Now note that any symmetry of methane must permute the hydrogen atoms. This gives rise to a permutation model of the symmetry group inside $S_4$; that is, the symmetry group of methane is isomorphic to a particular subgroup of $S_4$. Which subgroup?

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