Math 360, Fall 2015, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Transposition (or swap).
2. Even permutation.
3. Odd permutation.
4. Sign (of a permutation).
5. Alternating group.
6. Left congruence (on a group $G$ with respect to a subgroup $H$).
7. Left coset (of $H$ by $x$).
8. Right congruence.
9. Right coset.
10. Normal subgroup.

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

1. Theorem concerning generation of $S_n$ by cycles.
2. Theorem concerning generation of $S_n$ by transpositions.
3. Theorem concerning the sign of a permutation.
4. Theorem concerning the order of the alternating group.

Solve the following problems:

1. Section 9, problems 11 and 24.
2. Section 10, problems 1, 3, 6, 20, 21, 22, 23, and 24.
3. Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that $H$ is normal if and only if, for every $x\in G$ and every $h\in H$, the element $xhx^{-1}$ also lies in $H$. (In this case one says that $H$ is closed under conjugation by elements of $G$.)
4. In class we showed that the subgroup $\langle(12)\rangle\leq S_3$ is not normal in $S_3$. Show by explicit example that it violates the criterion you proved above.

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