# LOG#084. Group theory (IV).

Today we are going to speak about two broad and main topics: cyclic groups and some general features of finite groups (a few additional properties and  theorems).

A cyclic group is, informally speaking, a group that can be generated by a single element, the group has an element $g$ that we can call “the generator” of the group and we can “make” every element of the group by direct multiplication or “powering” the element g (or by multiplication by g once and once and again, in an additive language).

Definition (17). Cyclic group. We say a group $(G,\circ)$ is cyclic if

$\exists x\in G/G=\left\{e,x,x^2,\ldots,x^{n-1}\right\}$

The above group is at most a group with n-elements, and x is said to be “the generator” of the cyclic group.

Propostition. If $(G,\circ)$ is cyclic, then G is abelian.

Check: Let $a\in G$ with $a=x^p$ and let $b\in G=x^q$. Then

$a\circ b=x^p\circ x^q=x^{p+q}=x^{q+p}=x^q\circ x^p=ba$ QED

By the way, the product symbol/binary operation will be omited with frequency when no confusion in the composition of elements is possible.

Cyclic groups some have beautifuld and interesting features:

1st. The fundamental theorem of cyclic groups asis that every subgroup of a cyclic group is cyclic.

2nd. The order of any subgroup of a finite cyclic group of n elements is a divisor of n.

3rd. For every positive divisor k of n the group G has exactly one subgroup of order k.

This last property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor $d$ of $n$, the group has at most one subgroup of order $d$.

Proposition. Let G be a finite group, given $x\in G$ exists $n\in\mathbb{N}/x^n=e$.

Check: Given $x\in G$ if there is not a $n\in \mathbb{n}$ such that verifies that property, then there are some numbers $m', n' \in \mathbb{N}$ such that $x^{n'}=x^{m'}$. If not, the group is NOT finite! Therefore, if $n=n'-m'$, then we have

$x^n=x^{n'}x^{-m'}=e$

That is a contradiction and the proposition truth follows in a straightforward manner.

Proposition. Let G be a finite group. Let us define

$x\in G,n=\left\{\mbox{min}(m)/x^m=e\right\}$

and

$A_x=\left\{e,x^{n-1},\ldots,x\right\}$

Then, every element of $A_x$ is different.

Check: It there are 2 elements $x^p=x^q$ with $p,q then $x^{p-q}=e$ and $p-q, so n is not the minimum.

These features allow the following definition:

Definition (18). Given a finite group, and the element $x\in G$, the order of x is defined to be

$\mbox{order}(x)=\mbox{min}\left\{m/x^m=e\right\}$

Theorem (Lagrange). Let G be a finite group, then $\forall x\in G$ we have

$\dfrac{\vert G\vert}{\mbox{order}(x)}=n\in \mathbb{N}$

i.e., the natural number $\mbox{order}(x)$ is a divisor of $\vert G\vert$, the number of elements of G.

Proposition. Let $H\subseteq G$, $(H,\circ)$ be a subgroup of $(G,\circ)$, then the number of elements of H, $\vert H\vert$ is a divisor of $\vert G\vert$.

Some consequences of the Lagrange theorem are the following propositions:

Proposition. If $(G,\circ)$ is a group, and $\vert G\vert =p$, where $p$ is a prime number, then G is a cyclic group.

Proposition. For all $n\in \mathbb{N}$ exists, at least, the cyclic group with $n$ elements.

We will write from now $(G,\circ)$ as G for shorthand notation.

Definition (19). Conjugate elements and classes of conjugacy.

Let $x,y\in G$ two elements in G. They are said to be conjugated $x\sim y$ if exists an element $h\in G$ such as

$x=hyh^{-1}$

This definition provides an equivalence relation in G. The equivalence classes $\sim$ of conjugate elements in a group G are called “conjugacy classes” of G.

Definition (20). Conjugate elements for a subgroup. If G is a group and H is a subgroup of G, we ay that two elements $x,y\in G$ are related through H, and we write $x\equiv y(H)$ if $x^{-1}y\in H$.

Proposition. Let G a group and H a subgroup of G. Then,

a) The relation of conjugation defined above is an equivalence relation.

b) The equivalence class of $x\in G$ in this relation are in correspondence with the set $xH$, the product of elements of G by those in H.

Proposition. Let G be a group and H a subgroup. Then given an element $x\in G/x\ni H$, the set

$A=xHx^{-1}=\left\{xhx^{-1}/h\in H\right\}$

is a subgroup of G.

Definition (21). Conjugate group. Given $H\subseteq G$ a subgroup of G, and an element $x\in G$ then the subgroup

$A=xHx^{-1}$

is called the conjugate subgroup of H. Alternatively, A is conjugated with H if and only if (iff)

$\exists x\in G/xH=Ax$

See you in the next group theory blog post.