Math 360, Fall 2016, Assignment 10

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

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

1. Left coset (of a subgroup $H\leq G$ by an element $a\in G$).
2. Right coset.
3. $G/H$ (as a set).
4. Normal subgroup.
5. Coset multiplication.
6. $G/H$ (as a group).
7. Index (of a subgroup; this is Definition 10.13 in the text).

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

1. Description of the elements of the left coset $aH$.
2. Description of the elements of the right coset $Ha$.
3. Theorem concerning the cardinality of the various cosets of $H$.
4. Lagrange's Theorem.
5. Theorem concerning groups of prime order.
6. Theorem characterizing when coset multiplication is well-defined.
7. Theorem characterizing when $G/H$ is a group under coset multiplication.
8. Criteria for normality.

Solve the following problems:

1. Section 14, problems 1 and 9.
2. Suppose that $G$ is a group, and $H$ is a subgroup of $G$ with the property that, for any $a\in G$, $aH = Ha$. Prove that $H$ is a normal subgroup of $G$.
3. Let $G$ be any group, and suppose that $H$ is a subgroup of $G$ with index two (see Definition 10.13). Prove that $H$ is normal. (Hint: use the previous exercise. There are only two left cosets, namely $eH$ and something else. Similarly, there are only two right cosets, namely $He$ and something else. How are $eH$ and $He$ related? Therefore, how are the "something elses" related?)
4. Consider the dihedral group $D_n$, and let $R_n$ be the subgroup consisting of all rotational symmetries of the $n$-gon. Show that $R_n$ is a normal subgroup of $D_n$. (Hint: use the previous exercise.)
5. Find a familiar group isomorphic to the quotient group $D_n/R_n$. (Hint: what is the order of the quotient group?)

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