Math 120A HW 8

Due Wednesday, March 4.

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  1. Consider the group $G = \mathbb{Z}_9$ (under addition modulo $9$) and the cyclic subgroup $H = \langle 3 \rangle$. The group is abelian, so we can just talk about cosets rather than left or right cosets.
    1. Compute all of the cosets of $H$ in $G$ (for example, write $2+H = \{2,5,8\}$.)
    2. Which cosets in part (a) are equal to which other cosets in part (a)?
  2. Let $G$ be the group of all nonzero real numbers under multiplication and let $H$ be the cyclic subgroup of $G$ generated by $-2$. Let $a = 1$, $b = 1/2$, and $c = 1/3$. (Again, the group is abelian, so we can just talk about cosets rather than left or right cosets.)
    1. "Compute" the cosets $aH$, $bH$, and $cH$ (for example, write $aH = \{...\text{some elements}...\}$ where you list enough elements that the pattern is clear.)
    2. Let $d = -4$ and prove that $dH$ is equal to one of the cosets from part (a). (Give an actual proof, don't just say that they look the same so far.)
  3. Let $G$ be a group and let $H$ be a subgroup of $G$. Recall that the left coset relation $\sim_\text{L}$ on $G$ is defined by $a \sim_\text{L} b \iff b \in aH$. Prove that the left coset relation is transitive. (In class we proved that it is reflective and symmetric, so you can then conclude that it is an equivalence relation.) Hint: $b \in aH$ means $b = ah$ for some element $h \in H$.
  4. Let $G$ be a group and let $H$ be a subgroup of $G$.
    1. Assuming that $G$ is finite, prove that the number of left cosets of $H$ in $G$ is equal to the number of right cosets of $H$ in $G$ by using a counting argument.
    2. Explain why your counting argument for part (a) does not work when $G$ and $H$ are infinite.
    3. Prove in general (not assuming that $G$ is finite) that the number of left cosets of $H$ in $G$ is equal to the number of right cosets of $H$ in $G$. In particular, prove that there is a bijection $f$ from the set of all left cosets of $H$ in $G$ to the set of all right cosets of $H$ in $G$, given by $f(aH) = Ha^{-1}$. (Don't forget to prove that $f$ is well-defined! If we did the more obvious thing and wrote $f(aH) = Ha$, it would not define a function, because $aH = bH$ does not generally imply $Ha = Hb$.)
  5. Consider the group $\mathbb{Z}_3 \times \mathbb{Z}_3$ (with the operation of addition mod 3 in each coordinate.)
    1. Compute the cyclic subgroup generated by each element.
    2. Is $\mathbb{Z}_3 \times \mathbb{Z}_3$ cyclic? How could you answer this part without doing part (a) first?
    3. Prove that if $p$ is a prime number, then every element of $\mathbb{Z}_p \times \mathbb{Z}_p$ has order $1$ or $p$.
  6. Prove that the group $\mathbb{Z} \times \mathbb{Z}_2$ (with the operation of ordinary addition in the first coordinate and addition mod 2 in the second coordinate) is not cyclic. (Note that it doesn't make any sense to talk about a number being relatively prime to infinity, so you will need a different argument.)