Math 360, Fall 2020, 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

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

1. Normal subgroup.
2. $aHa^{-1}$ (the conjugate of the subgroup $H$ by the element $a$).
3. $G/H$ ("$G$ mod $H$," the left coset space associated with the pair $(G,H)$).
4. $H\backslash G$ (the right coset space associated with the pair $(G,H)$).

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

1. Criterion for equality of left cosets ("$aH=bH$ if and only if...")
2. Criterion for equality of right cosets (we did not discuss this in class, but it is straightforward to mimic the reasoning used for left cosets).
3. Theorem relating the cardinalities of the left cosets of $H$ to one another.
4. Theorem of Lagrange.
5. Classification of groups of prime order.
6. Theorem giving several conditions equivalent to normality of a subgroup.

Solve the following problems:

1. Section 10, problems 20, 21, 22, 23, 24, and 34.
2. Suppose that $G$ is abelian, and $H$ is any subgroup of $G$. Prove that $H$ is a normal subgroup of $G$.
3. For any group $G$, prove that the trivial subgroup $\{e\}$ is always normal.
4. For any group $G$, prove that the improper subgroup $G$ is always normal.
5. Suppose $G$ is any group, and $H$ is any subgroup of index two (i.e. for which there are exactly two left cosets). Show that $H$ is normal. (Hint: first show that the two left cosets are precisely (i) $H$ itself, and (ii) the set of elements of $G$ that do not belong to $H$. Then show the same for the right cosets.)
6. Working in $S_3$, compute all conjugates of the subgroup $\left\langle(12)\right\rangle$. Do you notice any patterns in your answer?
7. Working in $S_3$, compute all conjugates of the subgroup $\left\langle(123)\right\rangle$.
8. Try to find a one-line argument that predicts the answer to the last problem without any calculations. (Hint: what is the index of the subgroup?)
9. Let $G$ be any finite group with identity $e$, let $a$ be any element of $G$, and let $\left\lvert G\right\rvert$ denote the order of $G$. Show that $a^{\left\lvert G\right\rvert}=e$. This problem is not quite trivial, but it is well within your ability, and it is worthwhile to do it: it is the mathematical linchpin of the RSA cryptosystem, and for this and many other reasons we will make repeated use of it in the spring. Here is a hint: think carefully about the cyclic subgroup $\left\langle a\right\rangle$, and then combine Lagrange's Theorem with the laws of exponents.

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