LOG#083. Group Theory (III).

Posted: 2013/04/13 | Author: amarashiki | Filed under: Group Theory: basics, Physmatics | Tags: cayley table, center, finite group, group theory, order, subgroup | Leave a comment

Today we are going to study some interesting aspects of group theory.

Definition (14). Subgroup. Given a group $G$ and $H$ a nonempty subset of G, then H is said to be a subgroup of $G$ if and only if

$x\circ y^{-1}\in H,\forall x,y\in H$

Check: If $H$ is a subgroup, then we have

$x\circ y^{-1}\in H\forall x,y\in H$

This is evident, becuase if H is a subgroup, then $x,y\in H$ and $y^{-1}\in H$ and $\forall s,t \in H s\circ t\in H$ .

By the other hand, if $x\circ y^{-1}\in H$ $\forall x,y \in H$ then H is a subgroup.

And furthermore, it can be showed that associativy, neutral elerment and the inverse element in the subgroup are well-defined.

Definition (15). Finite group. A finite group $(G,\circ)$ is a group where its order $\vert G\vert$ is finite, i.e., if $\vert G\vert <\infty$ then the group is finite.

Definition (16). Cayley table. The Cayley table is an arrangement if form of “table” where a finite group “writes” its products as a “multiplication table”.

The Cayley table is a very useful device describing completely the structure of a finite group by arranging all the possible products of every group element in a square table similar of an addition or multiplication table. Many properties and features of a finite group can be easily “read-off” from this table by observing its structure:

a) A group is abelian if and only if the multiplication table is symmetrical along its diagonal axis (from left to right).

b) It can be read-off which elements are inverses of which elements by inspection.

c) The size and contents of the so-called group’s center.

d) The direct verification of the associativy property, something related to the so-called “Light’s associativity test”.

e) Due to the cancelation property, valid for groups, no row or column of a Cayley table may contain the same element TWICE. Thus each row and column of the table is a permutation of every element of the group. This fact limits which Cayley tables could be defined and which group operations are “valid”.

f) The distribution of identity elements on the Cayley table will be symmetric across the table’s main diagonal. Those that lie on the diagonal are their own inverse elements. This follows from the fact that in any group, even a non-abelian group, every element commutes with its own inverse element.

The order of the rows and columns of a Cayley table is in fact arbitrary. However, it is convenient to order them in the following manner: begin with the group’s identity element (always it is its own inverse), then list all the elements that are their own inverse elements, folowed by pairs of inverses listed adjacent to each other.

Remark: the center of a group G, denoted $Z(G)$ , is the set of elements that commute with every element of G.

$Z(G)=\left\{z\in G/\forall g\in G,zg=gz\right\}$

The center of G is always a subgroup of G.

See you in the next group theory blog post!

LOG#082. Group Theory (II).

Posted: 2013/04/13 | Author: amarashiki | Filed under: Group Theory: basics, Physmatics | Tags: abelian group, arrow, binary operation, category, category theory, equivalence class, function, functor, group axioms, map, order, quasigroup, semigroup, set | Leave a comment

Basic definitions of group theory: that is the topic today! We need some background previous to the “group axioms”.

Definition (1). Set is a collection of objects with some properties. Objects in the set are called “elements” or “members” of the set.

Definition (2). Category is some collection of objects linked or related theirselves with “arrows” or “maps”. Example: set is a category formed with sets as objects and functions as arrows. Other categories are, e.g., Top (topological spaces) or Ring (ring structures).

Category theory studies the main mathematical entities and their relationships, and it is a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. The nice fact about “group theory” is that there is a complete list of “group-like” structures: magma, semigroup, monoid, group, abelian group, loop, quasigroup, groupoid, category, semicategory, vector space, ring, semi-ring, and many other cool structures.

Definition (3). Function f is every map/relation/application between two sets X (domain) and Y(codomain) which associates each $x\in X$ with exactly one $y\in Y$ , and it is usally denoted by $f(x)$ . Mathematically speaking:

$f: X\longrightarrow Y$

Defintion (4). Functor F is every morphism/map/application of any category A to another category B.

$f:A\longrightarrow B$

Definition (5). Binary operation. Let $S\neq\emptyset$ be a set. A binary operation $\circ$ is any function taking “pairs” or “couples” of elements from that set into the own set. Mathematically speaking:

$\circ: S\times S\longrightarrow S$

and

$\circ(x,y)=x\circ y,\forall x,y\in S$

Examples of binary operations are the usual operations of addition(+), substraction(-), product(x) and division(/) of real numbers.

Definition (6). Given a set $X\neq \emptyset$ , a binary relation R in X is a subset $R\subset X\times X$ . Usually we write $xRy$ instead of the confusing notation $(x,y)$ .

Definition (7). A binary relation R in $X\neq \emptyset$ is called an equivalence relation if it satisfies the following conditions:

1) Reflexitivity: $xRx,\forall x\in X$ .

2) Symmetry: If $xRy$ then $yRx$ , $\forall x,y\in X$ .

3) Transitivity: If $xRy$ and $yRz$ , then $xRz$ , $\forall x,y,z\in X$ .

This gadget, the equivalence relation, allow us to classify the elements of every set in disjoint classes: the so-called equivalence classes.

Definition (8). Equivalence class. Let $R$ be an equivalence realtion, $a\in A$ is its equivalence class, denoted by $\left[a\right]$ , whenever we define

$\left[a\right]=\left\{b\in A/aRb\right\}$

We are ready for studying the group axioms right now!

Definition (9). Group is a pair $(G,\circ)$ where G is a non empty set ( $G\neq \emptyset$ ) and $\circ$ is an internal, binary operation in G that satisfy 3 simple properties:

1. Associativity: $x\circ(y\circ z)=(x\circ y)\circ z$ , $\forall x,y,z \in G$ .

2. Identity/neutral element ( $e$ ): $\exists e\in G/,$ it satisfies $e\circ x=x\circ e=x$ $\forall x\in G$ .

3. Invertibility/inverse element ( $x'=x^{-1}$ ): $\forall x\in G, \exists x'\in G/x'\circ x=x\circ x'=e$ .

That is all folks! Well, not exactly…There are other complementary concepts we can define:

Definition (10). Abelian group. An abelian group is any group $(G,\circ)$ with satisfy the commutativity property for its binary/product operation. That is, an abelian group satisfy that

$x\circ y=y\circ x$ $\forall x,y \in G$ .

Otherwise, the group is said to be non-abelian. Groups are generally non-abelian but there are some important abelian groups.

Definition (11). Semigroup. Let $G\neq \emptyset$ be a set, and $\circ: G\times G\longrightarrow G$ a binary operation that satisfies tha associative property. Then the pair $(G,\circ)$ is called a semigroup. That is, a semigroup is an associative binary operation but without the identity or inverse properties that groups have!

Definition (12). Quasigroup. A quasigroup is a pair $(G,\circ)$ such as $G\neq \emptyset$ is a set and the binary operation $\circ: G\times G\longrightarrow G$ satisfies the quasigroup law/property:

$\forall x,y\in G,\exists z,w\in G/x\circ z=y,\;\; w\circ x=y$

Definition (13). Order. The order of a group $(G,\circ)$ is the cardinal or number of elements in the structure $(G,\circ)$ .

Generally, the order of a group is denoted between vertical bars, $\vert G\vert$ or by $\mbox{ord}(G)$ .

If $(G,\circ)$ is a group, then its elements satisfy the following rules that can be derived from the definitions above in a trivial fashion:

1) The identity element $e$ , or neutral element, in a group is unique.

Check: If $e'\in G$ is another identity element, then $e=e\circ e'=e$ .

2) In a group G, if $x^{-1}$ is the inverse of $x\in G$ then $x^{-1}$ is the only inverse for that element.

The proof is equivalent to the previous one. It is let as an exercise for the reader.

3) $x\circ y=x\circ z\longrightarrow y=z,y\circ x=z\circ x\longrightarrow y=z\forall x,y,z\in G$ .

4) $x\circ y=e\longrightarrow x=y'\longrightarrow y=x',\forall x,y\in G$ , and where the prime denotes the inverse element.

5) $(x\circ y)'=y'\circ x'$ $\forall x,y\in G$ .

6) $(x')'=x\forall x\in G$ .

7) $(G,\circ)$ is abelian if and only if (iff) $(x\circ y)'=y'\circ x'=x'\circ y'$ $\forall x,y\in G$ .

See you in the next group theory blog post!

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