MAT102 Lecture 21B

• $p\ge 5$ is prime
• Show that $p\equiv 1,5(\mathrm{mod}6)$
• By division algorithm
• $p\equiv a(\mathrm{mod}6)$
• See that $p=6k+a$
• $p=6k$
  • $6\nmid p$
• $p=6k+1$
• $p=6k+2$
  • $6\nmid p-2$
• $p=6k+3$
• $p=6k+4$
• $p=6k+5$
• Only $p=6k+1$ and $p=6k+5$
• Question 3
  • $ca\equiv cb(\mathrm{mod}n)⟹a\equiv b(\mathrm{mod}\frac{n}{d})$
  • $n\mid ca-cb$ so $n\mid c(a-b)$
  • If $d=gcd(c,n)$
  • $dr\mid ds(a-b)$
  • By part 1
  • $r\mid s(a-b)$
  • By part 2
  • $r\mid a-b$
• Group, $(G,\oplus )$ is a set $G$ and an operator $\oplus :G\times G\to G$ such that something is true.
• Binary operator
• Must be associative, identity and inverses.
• $({\mathbb{Z}}_n,+)$
• $({\mathbb{Z}}_4,\cdot )$
• Is this a group
  • Identity
  • $[0{]}_4,[2{]}_4$ do not have inverses under multiplication
• ${\mathbb{Z}}_4=\{[0],[1],[2],[3]\}$
• ${\mathbb{Z}}_4^{\times }=\{[1],[3]\}$
  • This is a group under multiplication
  • Closure
• If we take ${\mathbb{Z}}_n^{\times }$ will always be a group under multiplication #tk
• 1
  • ${\mathbb{Z}}^{+}$ is a group under addition?
  • Associative
    • $(a+b)+c$
    • $a+(b+c)$
  • Identity
    • There is no identity, $0$ is not in ${\mathbb{Z}}^{+}$
  • What is ${\mathbb{Z}}^{\ast }$
• 2
  • $r$ is a rotation $120$ degrees right
  • $s$ is a reflection
  • Read right to left
  • $r(\mathrm{△})=$ the rotation of the triangle
  • $sr(\mathrm{△})=s(r(\mathrm{△}))$
  • init: ACB
  • BCA
  • CBA
  • CAB
  • BCA
  • BAC
  • $D_n=\{e,r,r^2,\dots ,r^{n-1},s{,}^2,s{r}^2,\dots ,s{r}^{n-1}\}$
  • Abelian means it's commutative
  • See that $s^2{r}^3=e$ so this is not abelian
• 3
  • $S_5$
  • $\sigma =(13)(235)(13)(42)$
  • What's a bijection
  • If we have $X=\{1,2,3,4,5\}$
  • $S_5=\{f:X\to X,f\text{ bijection}\}$
  • $(235)(n)=\{\begin{array}{ll}1& n=1\\ 3& n=2\\ 5& n=3\\ 4& n=4\\ 2& n=5\end{array}$
  • This is how you define it.
  • $\sigma =(13)(235)(13)(42)$
  • If we start with $n=2$
  • $\sigma =(13)(235)(13)(42)=(2415)(3)$
  • Because $(3)$ is the identity map, then it doesn't mater
  • $(4152)=(1524)=(5241)$
  • Same cycle
  • Also not abelian