Math 360, Fall 2016, Assignment 9

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

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

1. Even permutation.
2. Odd permutation.
3. Sign (of a permutation).
4. Alternating group.
5. Dihedral group.
6. Left congruence (modulo a subgroup $H$).

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

1. Theorem concerning the number of transpositions appearing in a decomposition of a given permutation into transpositions.
2. Rules of sign for permutations.
3. Formula for the inverse of a product.
4. Theorem characterizing left congruence as a relation of a certain type.

Solve the following problems:

1. Let $G$ be any group and let $H$ be a subgroup of $G$. Define a relation $\sim_{r,H}$ (called "right congruence modulo $H$") by declaring that $g_1\sim_{r,H}g_2$ if and only if $g_2g_1^{-1}\in H$. Prove that right congruence is always an equivalence relation on $G$. (Note that this is at least potentially different from our definition of "left congruence," in which $g_1\sim_{l,H}g_2$ if and only if $g_2^{-1}g_1\in H$. The reason for calling these relations "left" and "right" congruence will become clear next week.)
2. Prove that if $G$ is abelian, then left and right congruence are the same relation (i.e. $g_1\sim_{l,H}g_2$ if and only if $g_1\sim_{r,H}g_2$).
3. Section 10, problems 1, 3, 6, and 7. (Note: the "left coset" of a group element means its equivalence class with respect to left congruence. The "right coset" means the equivalence class with respect to right congruence. When $G$ is abelian, these two relations coincide, so it is harmless to speak simply of the "coset" of the group element.)

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