Math 120A HW 6

Due Tuesday, November 25.
  1. Let $G$ be an abelian group, let $H$ be a subgroup of $G$, and let $a,b \in G$. Prove that if $a \in H + b$ then $H + a = H + b$. (This is part of a proof of a theorem from class; I decided not to assign the other part.) Hint: how do you use the hypothesis $a \in H + b$? Write explicitly what it means. Then show that $H + a \subseteq H + b$ and vice versa.
  2. Note that $15\mathbb{Z}$ and $5\mathbb{Z}$ are both subgroups of $\mathbb{Z}$, and $15\mathbb{Z}$ is a subset of $5 \mathbb{Z}$, so $15\mathbb{Z}$ is a subgroup of $5\mathbb{Z}$ (you don't have to prove this part.) Find all of the cosets of $15\mathbb{Z}$ in $5\mathbb{Z}$. How many are there? (Make sure you use the definition of "coset" with additive notation, not multiplicative notation.)
  3. Let $G$ be a group, let $H$ be a subgroup of $G$, and let $K$ be a subgroup of $H$ (so $K$ is also a subgroup of $G$.)
    1. Prove that if $G$ is finite, then $(G : H)(H : K) = (G : K)$.
    2. This equation is true even without the finiteness assumption (although I won't ask you to prove it.) How can we use this equation to calculate the number of cosets of $15\mathbb{Z}$ in $5\mathbb{Z}$? Hint: you can use the fact that the number of cosets of $n \mathbb{Z}$ in $\mathbb{Z}$ is $n$.
  4. Consider the cyclic subgroup $H$ of $S_3$ generated by the cycle $(1\;2\;3)$.
    1. What is the index of $H$ in $S_3$? Hint: because $S_3$ is finite you don't have to find the cosets of $H$ yet; you can just do a calculation with cardinalities.
    2. Find all of the left cosets of $H$ in $S_3$.
    3. Find all of the right cosets of $H$ in $S_3$.
    4. Prove that if $G$ is any group, $H$ is any subgroup of $G$, and $H$ has index $2$ in $G$, then $aH = Ha$ for all $a \in G$. Hint: the left cosets form a partition of $G$, and the right cosets also form a partition of $G$. What is special about the number 2 here?
  5. List all of the subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_4$ and draw them in a subgroup diagram. Hint: In general, if $H_1$ is a subgroup of $G_1$ and $H_2$ is a subgroup of $G_2$, then $H_1 \times H_2$ is a subgroup of $G_1 \times G_2$. But $G_1 \times G_2$ may have other subgroups not of this form.
  6. Prove that the groups $\mathbb{Z}_2 \times \mathbb{Z}$ and $\mathbb{Z}_3 \times \mathbb{Z}$ are not isomorphic.

Optional practice problems from the book (these will not be graded): Section 10 Exercises 1, 3, 6, 7, 34. Section 11 Exercises 1, 5, 15.