Math 360, Fall 2017, Assignment 9

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

- Augustus de Morgan

Read:

1. Section 10.

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

1. Transposition.
2. Parity (of a permutation).
3. Sign (of a permutation).
4. $A_n$ (i.e. the alternating group).
5. $D_n$ (i.e. the dihedral group).
6. Left congruence (modulo a subgroup $H$).
7. $gH$ (i.e. the left coset of $H$ by $g$).

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

1. Theorem concerning generation of $S_n$ by transpositions.
2. Theorem comparing the number of orbits of $\pi$ to the number of orbits of $\tau\pi$ (where $\tau$ is a transposition).
3. Theorem concerning the numbers of transpositions in two different decompositions of $\pi$ as products of transpositions.
4. Formula for the order of the alternating group.
5. Formula for the order of the dihedral group.
6. Theorem describing the elements of $gH$.
7. Criterion for equality of cosets (i.e. saying when $g_1H=g_2H$).
8. Theorem concerning the sizes of cosets of $H$.
9. Lagrange's theorem.

Solve the following problems:

1. Section 10, problems 1, 3, 5, 6, 21, 22, 23, and 24.

--------------------End of assignment--------------------