LOG#084. Group theory (IV).

grouptheory4

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<n then x^{p-q}=e and p-q<n, 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.

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