Math 360, Fall 2019, 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. Normal subgroup.

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

1. Theorem describing the elements of $aH$.
2. Criterion for equality of left cosets ("$aH=bH$ if and only if...").
3. Theorem describing the elements of $Ha$.
4. Criterion for equality of right cosets ("$Ha=Hb$ if and only if...").
5. Theorem relating the cardinalities of the left cosets of $H$ to one another.
6. Theorem of Lagrange.
7. Classification of groups of prime order.
8. Theorem concerning the normality of subgroups, when $G$ is abelian.
9. Theorem concerning the normality of the trivial subgroup.
10. Theorem concerning the normality of the improper subgroup.

Solve the following problems:

1. Section 10, problems 1, 3, 6, 7, 12, 13, 15, 34, and 40.
2. Prove that any subgroup of index two is normal. (Hint: let $H$ be a subgroup of index two, and let $a\in G$ be any element not in $H$. Then the left cosets are $eH$ and $aH$, while the right cosets are $He$ and $Ha$. We observed in class that $eH=He=H$. So how are $aH$ and $Ha$ related?)
3. Prove that $A_n$ is a normal subgroup of $S_n$, for any $n$.

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