6.Cyclical group

definition

In group $<G,\ast >$, if there's an element $a\in G$, makes for $\mathrm{\forall }x\in G$, exists $i\in \mathbb{Z}$ satisfies:

$$x=a^i$$

then $G$ is a Cyclical group, denoted as $G=<a>$, and a is a generator of $G$. The set of all generators of $G$ is called the generator set of $G$​.

That means, every element of $G$ can be represented by an integer power of $a$​.

The set $S_a=\{a^n\mid n\in \mathbb{Z}\}$, formed by the integer powers of any arbitrary element $a$ in group $G$, is a subgroup of $G$. This subgroup is called the cyclic subgroup generated by $a$, and $a$ is called the generator of this cyclic subgroup.

Clearly, cyclic groups can be seen as a special case of cyclic subgroups, when $G=<a>$ is a cyclic group, i.e., $G=\{a^n\mid n\in \mathbb{Z}\}$​.

Typical cyclical group

(1) Integer addition group $<\mathbb{Z},+>$ is cyclic group of infinite order, it's generator group is $\{1,-1\}$.

(2) Integer module $n$ congruential class addition group $<{\mathbb{Z}}_n,\oplus >$ is a cyclic group of order $n$, it's generator set is $M=\{a|a\in {\mathbb{Z}}_n,(a,n)=1\}$, and number of generator is $\varphi (n)$; if $n$​ is a prime number, then all element are generator except identity element 0.

In fact, these are the only two cyclic groups.

(3) If $n$ is prime, then multiplicative group of integral modulo $n$ congruence class $<{\mathbb{Z}}_n^{\ast },\otimes >$ is a $n-1$​ order cyclical group.

Property

If $<G,\ast >$​ is cyclical group, then it's an Abel group.

Finite group $<G,\ast >$ is a cyclical group if and only if there's at least one element $a$ in $G$ makes $|a|=|G|$​.

Isomorphism of cyclic groups

Let $<G,\ast >=<a>$ be a cyclic group generated by $a$, then:

(1) If $G$ is an infinite set, then $G$ is isomorphic to the additive group of integers $<\mathbb{Z},+>$.

(2) If $|G|=n$, then $G$ is isomorphic to the additive group of integers modulo $n$, denoted as $<{\mathbb{Z}}_n,\oplus >$​.

Generator of a cyclic group

(1) The infinite cyclic group $G=<a>$ has exactly two generators.

(2) For the cyclic group of prime order $p$, $G=<a>$, every element except the identity is a generator of $G$.

(3) For the cyclic group of positive integer order $n$, $G=<a>$, for any $y=a^x\in G$, if $(n,x)=1$, then $y$ must be a generator of $G$​.

Subgroup of a cyclic group

Subgroup of cyclic group is also cyclic group, and

(1) All subgroup of $<\mathbb{Z},+>$ are $H_m=<m>,m=0,1,2,...$

(2) All subgroup of $<{\mathbb{Z}}_n,\oplus >$ are $<0>$ and $<d>$, $d|n(0<d<n)$.