Math 360, Fall 2020, Assignment 12

"When I think of Euclid even now, I have to wipe my sweaty brow."

- C. M. Bellman

Read:

1. Section 10.

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

1. $\sim_{H,l}$ (the relation of left congruence modulo $H$ on a group $G$).
2. $aH$ (the left coset of $H$ by $a$).
3. $(G:H)$ (the index of H in G; see section 10 of the text for this definition).
4. $\sim_{H,r}$ (the relation of right congruence modulo $H$ on a group $G$).
5. $Ha$ (the right coset of $H$ by $a$).
6. $(G:H)$ (the index of $H$ in $G$; see Definition 10.13 in the text).

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

1. Theorem describing the elements of $aH$.
2. Theorem describing the elements of $Ha$.

Solve the following problems:

1. Section 10, problems 1, 3, 6, 7, 12, 13, and 15.

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