Math 360, Fall 2021, Assignment 14

I must study politics and war, that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

Read:

1. Section 10.

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

1. $\sim_{l,H}$ (the relation of left congruence modulo the subgroup $H$).
2. $\sim_{r,H}$ (the relation of right congruence modulo the subgroup $H$).
3. $xH$ (the left coset of $H$ by $x$).
4. $Hx$ (the right coset of $H$ by $x$).
5. $(G:H)$ (the index of $H$ in $G$; note that we did not discuss this in class, but it is Definition 10.13 in the text).

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

1. Theorem concerning a special property of kernels, not shared by arbitrary subgroups.
2. Example to show that not every subgroup can be the kernel of a homomorphism.
3. Theorem concerning the properties of the left congruence relation ("$\sim_{l,H}$ is an...").
4. Theorem concerning the properties of the right congruence relation ("$\sim_{r,H}$ is an...").
5. Theorem describing the elements of $xH$.
6. Theorem describing the elements of $Hx$.
7. Theorem relating left and right congruence when $G$ is abelian.
8. Example to show that the result of the previous theorem may be false when $G$ is not abelian.
9. Theorem relating the sizes of $xH$ and $yH$.
10. Lagrange's Theorem.

Solve the following problems:

1. Section 10, problems 1, 3, 6, 7, 9, 10, 12, 15, 20, 21, 22, 23, and 24.

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