Math 360, Fall 2021, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Section 13.

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

1. Even permutation.
2. Odd permutation.
3. $A_n$ (the alternating group on $n$ letters).
4. Homomorphism.
5. Monomorphism.
6. Epimorphism.
7. Forward image (of a subset, under a function).
8. Pre-image (of a subset, under a function).
9. Pushforward (of a subgroup, under a homomorphism).
10. Pullback (of a subgroup, under a homomorphism).
11. Image (of a homomorphism).
12. Kernel (of a homomorphism).

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

1. Theorem concerning the orbit count of $\tau\pi$, where $\tau$ is a transposition and $\pi$ is an arbitrary permutation.
2. Theorem concerning the sign of $\tau\pi$, where $\tau$ is a transposition and $\pi$ is an arbitrary permutation.
3. Formula for $\mathrm{sgn}(\tau_1\dots\tau_k)$, where $\tau_1,\dots,\tau_k$ are transpositions.
4. Formula for $\mathrm{sgn}(\pi\sigma)$, where $\pi$ and $\sigma$ are arbitrary permutations.
5. Formula for the order of $A_n$.
6. Theorem characterizing the pushforward of a subgroup ("The pushforward of a subgroup is a...")
7. Theorem characterizing the pullback of a subgroup ("The pullback of a subgroup is a...")

Solve the following problems:

1. Section 13, problems 1, 2, 3, 8, 9, 10, 17, and 18.
2. Determine whether the following permutations are odd or even: (i) $(12)$, (ii) $(123)$, and (iii) $(1234)$.
3. Suppose $\sigma$ is an $l$-cycle, where $l$ is even. Is $\sigma$ an even permutation or an odd permutation?
4. Suppose $\sigma$ is an $l$-cycle, where $l$ is odd. Is $\sigma$ an even permutation or an odd permutation?
5. Define a function $f:\{1,2,3,4\}\rightarrow\{a,b,c,d\}$ as follows: $f(1)=a, f(2)=a, f(3)=b, f(4)=c$. Compute the following: (i) $f[f^{-1}[\{c,d\}]]$ and (ii) $f^{-1}[f[\{2,3\}]]$.
6. Suppose $f:A\rightarrow B$ is a function, and $S\subseteq A$. Prove that $f^{-1}[f[S]]\supseteq S$. Give examples to show that equality may or may not hold.
7. Suppose $f:A\rightarrow B$ is a function, and $T\subseteq B$. Prove that $f[f^{-1}[T]]\subseteq T$. Give examples to show that equality may or may not hold.

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