Math 360, Fall 2015, Assignment 9

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

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

1. Index (of a subgroup) (Note: we did not define this in class. You can find the definition on page 101 of the text.)
2. Direct product (of two groups).
3. Direct sum (of two abelian groups written additively).

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

1. Criteria for the equality of cosets.
2. Criteria for the normality of subgroups.
3. Lagrange's Theorem.
4. Theorem concerning groups of prime order.
5. Theorem concerning the "copies" of $G$ and $H$ inside $G\times H$.
6. Criterion for $G$ to be isomorphic to the direct product of two of its subgroups.

Solve the following problems:

1. Section 10, problems 13, 15, and 34.
2. Section 11, problems 1, 2, 3, and 8.
3. Is $S_3$ isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_3$? Prove your answer.
4. Working in $S_3$, consider the subgroups $H=\left\langle(12)\right\rangle$ and $K=\left\langle(123)\right\rangle$. Show that $H\cap K$ is trivial, and that $H\cup K$ generates all of $S_3.$
5. Are your answers to the previous two problems consistent with one another? Explain.

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