Math 360, Fall 2018, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Section 10.

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

1. Left congruence (with respect to a subgroup $H$ of a group $G$).
2. Right congruence (with respect to a subgroup $H$ of a group $G$).
3. $aH$ (the left coset of $H$ by $a$).
4. $Ha$ (the right coset of $H$ by $a$).
5. Normal subgroup.
6. Index (of a subgroup; we did not discuss this in class yet, but it is Definition 10.13 in the text).

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

1. Theorem relating the cardinality of the coset $aH$ to the cardinality of $H$.
2. Lagrange's theorem.
3. Corollary concerning groups of prime order.
4. Theorem giving four additional conditions equivalent to the condition that left and right congruence are the same relation.

Solve the following problems:

1. Section 10, problems 1, 3, 5, 6, 7, 12, 13, 15, 20, 21, 22, 23, 24, and 34.

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