Math 120A HW 3

1. Recall that $+_n$ and $\cdot_n$ denote the operations of addition modulo $n$ and multiplication modulo $n$ respectively on the set $\mathbb{Z}_n = \{0,1,\ldots,n-1\}$. They are computed by adding or multiplying normally and then reducing modulo $n$.
   1. Write the table of the operation $+_4$ on $\mathbb{Z}_4$. Does the structure $(\mathbb{Z}_4,+_4)$ have an identity element, and if so, what is it? Which of its elements have inverses? For each invertible element, say what its inverse is.
   2. Write the table of the operation $\cdot_4$ on $\mathbb{Z}_4$. Does the structure $(\mathbb{Z}_4,\cdot_4)$ have an identity element, and if so, what is it? Which of its elements have inverses? For each invertible element, say what its inverse is.
2. For each of the following objects, say whether or not it is a group. If it is not a group, say why not. (Hint: first you should check that the object is a binary structure. Also, you should check that it has an identity element before considering the existence of inverses.)
   1. $(\mathbb{R}, \max)$ where $\max(a,b)$ denotes the maximum of two real numbers $a$ and $b$.
   2. $([0,1],\max)$ where $[0,1] = \{x \in \mathbb{R} : x \ge 0 \text{ and } x \le 1\}$.
   3. $(\mathbb{R}^*,+)$ where $\mathbb{R}^*$ denotes the set of all nonzero real numbers and $+$ denotes ordinary addition.
3. Recall that isomorphisms of binary structures respect identity elements. In this problem, we show that homomorphisms of groups respect identity elements. (Although they are related, neither statement is an obvious consequence of the other.)
   1. Show that in any group, the identity element is the unique solution to the equation $x \ast x = x$.
   2. Use this to show that if $(G,\ast)$ and $(G',\ast')$ are groups and $\phi$ is a homomorphism from $(G,\ast)$ to $(G',\ast')$, then $\phi(e_G) = e_{G'}$.
4. Let $(G,\ast)$ be a three-element group, say $G = \{e,a,b\}$ where $e$ is the identity element. Here is a partly filled-in group table for $(G,\ast)$: \begin{array}{c|ccc} \ast & e & a & b\\ \hline e & e & a & b\\ a & a & ? & \\ b & b & & \end{array}
   1. What goes in the space indicated with a question mark? Explain why there is only one possibility.
   2. Fill in the rest of the table.
   (This problem shows that, up to isomorphism, there is only one three-element group.)
5. Consider the group $(S,\cdot)$ where $$ S = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \right \} $$ and $\cdot$ denotes matrix multiplication. (You may assume that this structure is a group.) Prove that $(S,\cdot)$ is not isomorphic to $(\mathbb{Z}_4, +_4)$. Hint: what happens when you compute $a +_4 a$ for $a \in \mathbb{Z}_4$? What happens when you compute $a \cdot a$ for $a \in S$?