chapter 3 summary

# Definitions

Cyclic group
A group $G$ is cyclic if there exists $a\in G$ such that $\langle a \rangle = G.$

The Order of A
The smallest positive integer $n$ such that $a^n = e$ . If there is no such integer we say $a$ has infinite order, we write $\mid a \mid = n$. Note that: $\mid a \mid = \mid\langle a \rangle\mid$ .

# Theorems

Theorem 3.1
Let $a\in G$ with $G$ as a group.
Then $\langle a \rangle:=\{a^k:k\in \Bbb{Z}\}$ is a subgroup of $G$. (or $\{ka:k\in \Bbb{Z}\}$ if operation is addition)
Furthermore, $\langle a \rangle$ is the smallest subgroup containing $a$.

Theorem 3.2
Every cyclic group is abelian.

Theorem 3.3
Every subgroup of a cyclic group is cyclic.

Corollary 3.4
The subgroups of $\Bbb{Z}$ are $n\Bbb{Z}$ where $n=0,1,2, ...$

Proposition 3.5
Let G be a cyclic group of order n and suppose that $a$ is
a generator for G. Then $a^{k} = e$ if and only if n divides k.

Theorem 3.6
Let $G$ be a cyclic group of order n and suppose that $a\in G$
is a generator of the group. If $b = a^k$, then the order of b is $n/d$, where
$d = gcd(k, n)$.

Corollary 3.7
The generators of $\Bbb{Z}_n$ are the integers $r$ such that $1\leq r<n$ and gcd$(r,n)=1$

# Examples & Notes

Examples of abelian's and cyclics

$D_3$ has 6 generators, is not abelian and not cyclic
$V_4$ is abelian and not cyclic
$Q_8$ is not abelian and not cyclic
$D_4$ has 8 generators, is not abelian and not cyclic
$\Bbb{Z}$ is abelian and cyclic and is a group of infinite order
$V_4$ the symetry group for a rectangle is cyclic but not abelian. An example of a smallest group from $V_$$is$<h>$=$\{{e,h}\}$where$h$represents a horizontal flip. The order of$<h>$or$\mid<h>\mid$= 2 The converse of theorem 3.2 is false The statement: every abelian group is cyclic is false counter examples;$V_4$&$U(8)$3.37 Prove that if$G$has no proper nontrivial subgroups, then$G$is cyclic groups. Proof: If$g \in G$where$g \neq e$&$<g> = G$is a proper subgroup of$G$. Therefore,$g$is a generator of$G$and thus,$G$is cyclic.$qed\$

page revision: 24, last edited: 19 Apr 2010 22:16