Math 120A HW 3
Due Tuesday, October 28.
- An automorphism of a group is an isomorphism of that group with itself.
- List all automorphisms of the group $\mathbb{Z}_4$.
- Give a bijection $f$ of $\mathbb{Z}_4$ with itself such that $f(0) = 0$, but $f$ is not an automorphism of $\mathbb{Z}_4$.
Justify your answers.
- Let $(G,\ast)$ be a group and let $a \in G$. We write $a^n$ for $a \ast a \ast \cdots \ast a$ where there are $n$ copies of $a$. (There is no need to write parentheses because the operation is associative.) Prove that if $a^5 = e$ and $a^3 = e$, then $a = e$. Hint: which powers of $a$ can you show are the identity?
- Let $U$ be a set. Prove that the binary structure $(\mathcal{P}(U), \triangle)$ is an abelian group, where $\mathcal{P}(U)$ denotes the power set of $U$ (the set of all subsets of $U$) and $\triangle$ denotes the operation of symmetric difference of sets, defined by
$$A \mathbin{\triangle} B = (A \cup B) - (A \cap B).$$
Hint for associativity: show that the conditions $x \in
(A \mathbin{\triangle} B) \mathbin{\triangle} C$ and $x \in
A \mathbin{\triangle} (B \mathbin{\triangle} C)$ are equivalent by
splitting into cases according to
how many of the three sets $A$, $B$, $C$ the element $x$ belongs to.
- Find all subgroups of $\mathbb{Z}_{10}$ and represent them in a subgroup diagram. You don't need to write a proof that your diagram is complete (although you should have a proof in mind if you want to be sure that you are correct.)
- Consider $\mathbb{Z}$ as a group under addition and let $a, b \in \mathbb{Z}$. Prove that $\langle a \rangle \subseteq \langle b \rangle$ if and only if $b \mid a$ (meaning $b$ divides $a$.)
- A group $G$ is defined to be cyclic if there is an element $a \in G$ such that $\langle a \rangle = G$. Prove that the following groups are NOT cyclic:
- $(\mathbb{Q}, +)$.
- $(\mathbb{Z} \times \mathbb{Z}, +)$ where the operation is defined coordinate-wise: $(a_1, a_2) + (b_1, b_2) = (a_1 + b_1, a_2 + b_2)$.
- Let $G$ be a group and let $a$ and $b$ be elements of $G$.
Assume that $a^5 = e$ and $b^3 = e$. Find the inverse of the element
$g = ab^2a^3ba^4$. Express $g^{-1}$ as a product of positive powers
of $a$ and $b$. Hint: here is how you would do a similar problem:
the inverse of $abc$ is $c^{-1}b^{-1}a^{-1}$ (why?) and if $a^4 = b^4 =
c^4 = e$ we would have $c^{-1} = c^3$, $b^{-1} = b^3$, and $a^{-1} = a^3$,
so we can write $(abc)^{-1} = c^3b^3a^3$ (then we can check that the definition
of inverse is satisfied.)
- Let $G$ be a group and let $a \in G$. Assume that $a^n \ne e$ for all positive integers $n$. Prove that the function $\phi: \mathbb{Z} \to \langle a \rangle$ defined by $\phi(i) = a^i$ is injective. (Together with what we did in class, this shows that $\phi$ is an isomorphism.)
- Find the orders of the elements 5, 6, and 7 in $\mathbb{Z}_{10}$.
Optional practice problems from the book (these will not be graded): Section 4 Exercises 17, 32, 34; Section 5 Exercises 23, 36, 51, Section 6 Exercises 19, 21, 44