Math 360, Fall 2018, Assignment 10

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

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. $S_n$.
3. Two-row notation (for permutations).
4. Order (of a group element, e.g. a permutation).
5. $(i_1,\dots,i_k)$ (i.e. the cycle determined by $i_1,\dots,i_k$).
6. Disjoint (cycles).

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

1. Formula for the order of $S_n$.
2. Theorem concerning the disjoint cycle decomposition of a permutation.
3. Formula for the order of a permutation (in terms of its cycle lengths).
4. Cayley's Theorem.

Solve the following problems:

1. Section 8, problems 1, 3, 5, 7, 9, 30, 31, 32, and 33.
2. Section 9, problems 7, 8, 9, 10, 11, 12 (in problems 10-12 simply find the disjoint cycle decomposition), 13, 14, 15, and 16.
3. (Proof of Cayley's Theorem) Let $G$ be any group. For any $g\in G$, define a function $l_g:G\rightarrow G$ by the formula $l_g(x)=gx$.

(i) Show that $l_g$ is injective.

(ii) Show that $l_g$ is surjective.

(iii) Show that $l_g\in\mathrm{Sym}(G)$.

(iv) Now define a map $\phi:G\rightarrow\mathrm{Sym}(G)$ by the formula $\phi(g)=l_g$. Show that $\phi(gh)=\phi(g)\circ\phi(h)$.

(v) Show that the map $\phi$ defined above is injective.

(vi) Show that the image of $\phi$ is a subgroup of $\mathrm{Sym}(G)$.

(vii) Show that $G$ is isomorphic to some subgroup of $\mathrm{Sym}(G)$.

4. Carry out all the steps of the proof above explicitly for $G=\mathbb{Z}_3$. Thus, find an explicit isomorphism between $\mathbb{Z}_3$ and some subgroup of $S_3$.

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