This work describes the stairss taken in the development of a computing machine application for analysing finite groups. It discusses all constructs related to the survey of finite Groups that are implemented in the computing machine application. Finite groups play a major function in the survey of many mathematical objects such a geometric forms and root of multinomials. The application allows for the survey of the undermentioned finite group structures ; commutator, Centre, conjugate categories, centralisers of elements of the group, normalisers of subgroups of a group. The application predefines four categories of groups for analysis, which are ; the Symmetric group on n letters, the Alternating group on n letters, the Dihedral group and the Cyclic group, it besides supports user defined groups specified as a Cayley tabular array.

## Keywords: Finite, Groups, Application, Alternating, Roots, Polynomials, Cyclic,

## Dihedral

## 1.1 Introduction

In this work the theory and constructs in abstract algebra and package technology rules are used in the development of an application for the analysis of predefined finite groups. Abstract Algebra in a broadest sense is the survey of sets and operations. One of the most of import of these sets with operations is called a group. Few constructs have penetrated mathematics every bit profoundly as the construct of group and group action ( Menini and Oystaeyen, 2004 ) .

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In mathematics, a group is an algebraic construction dwelling of a set together with an operation that combines any two of its elements to organize a 3rd component that is besides in the Groups set ( Fraleigh, 2003 ) . To measure up as a group, the set and the operation must fulfill a few conditions called group maxims, viz. closing, associativity, individuality and invertibility. Groups can be classified via assorted standards such as the cardinality of the implicit in set, nature of the binary operation and besides with regard to the being or non being of certain subgroups. With regard to the cardinality of the implicit in set a group is said to be finite if its underlying set is finite and is infinite otherwise, while on the groups binary operation a group can be classified as either abelian or non-abelian if the double star operator is commutative or non-commutative severally and eventually with the being of certain subgroups, a group can be classified as simple or non-simple if the group has a nontrivial normal Subgroup or does n’t severally.

Very of import categories of finite groups in an undergraduate class in abstract algebra include the Cyclic groups, Dihedral groups, and Symmetric groups. Of peculiar importance is the Symmetric group which has the of import belongings that every finite group of order Ns.

## 2.1 Methodology

In this work information was obtained utilizing the assorted system analysis and design constructs and techniques. Relevant information was obtained from ; text editions, theses, on-line publications, relevant web sites, audience with professionals in the field of group theory and Interaction with the intended users.

The system was designed utilizing J2SDK ( Java 2 Software Development Kit ) in a JCreatortm IDE. The category files for the full system are organized into a individual bundle and beginning codification into a individual booklet. Java nucleus and extension bundles were used to congratulate the system. The computing machine used to develop the system was a HP Pavillon dv2000 with Intel Centrino CPU, 1G RAM and Windows Vista operation system with jdk1.6.0_06 installed.

## 3.1 Result and Discussion

## 3.2 Consequence

The consequence of the work was an application that can analyse finite groups that are either predefined as portion the application or defined by the user. The application does non detect new groups but allows users to interact with the defined groups right from the elements of the groups up to the subgroups of the group, the application allows the user to change the representation of the groups elements and make new groups form the subgroups of bing groups.

The developed application supports the undermentioned categories of finite groups ; dihedral group of a regular n-gon ( where N is positive and less than or equal to two hundred and 50 ) , symmetric and jumping group on n letters ( where N is positive and less than or equal to six ) , cyclic group with n elements ( where N is positive and less than or equal to five hundred ) . The application can easy be extended to back up other categories of finite groups. When group is selected its elements are displayed in a list. A lexicographic ordination has been imposed on all group elements, therefore consequences that consist of group elements appear lexicographically.

The system consists of three functional constituents that are seen and interacted with by the user. These are ; the finite group choice and alteration console, finite group analysis console and the end product screen.

Group choice, definition and alteration console

This constituent is responsible for supplying an interface between the user and the groups supported by the system. It is used by the user to do Group choices, change the visual aspect of a group and make new Groups from the bing Groups. The execution of this constituent is outlined in Figure1.

The execution consists of a button and a jazz band box. The jazz band box contains the options ; symmetric group, jumping group, dihedral group, cyclic group, for stipulating a group in those categories of groups. The duologue for stipulating a group in the jumping group is shown in Figure 2. The duologue for the other categories of groups is similar to that shown in Figure 2.

The option “ subgroup of current group ” crates a group from the current active group. When this option is activated it checks the elements in the group selected by the user, if they form a subgroup so those selected elements becomes the new active group. The option “ alteration group position ” produces the duologue shown in Figure 3 for stipulating the new representation of the group elements. The concluding option “ input cayley tabular array ” produces two duologues shown in Figure 4 and Figure 5 for stipulating the cayley tabular array.

Figure 1: Interface for the choice, definition and alteration of groups

Figure 2: Dialogue for stipulating an jumping group

Figure 3: Dialogue for stipulating the new representation of the group elements

Figure 4: Dialogue for stipulating the elements in a cayley tabular array

Figure 5: Dialogue stipulating the entries of a cayley tabular array

Finite group analysis console

This constituent is responsible for supplying an interface for the survey of the group construction of groups selected by the group choice and alteration console. It provides functionality for the calculation and show of group constructions. The execution of this constituent is shown outlined in Figure7. The execution consists of seven buttons, six of which are used for group analysis while the last button is used to unclutter the end product screen.

Figure 7: interface for group analysis

The normal or subgroup ” button is used to find if a set of selected group elements organize a subgroup or normal subgroup. The buttons action is shown in Figure 8.

## Figure 8: Testing if a subset of a group forms a subgroup or normal subgroup

The “ normaliser ” button is used to calculate the normaliser of subgroup. The subgroup is specified by choosing each of the elements that form the subgroup. The action of the “ normaliser ” button is shown in Figure9

Figure 9: Calculation of the normalize of a subgroup

The “ cosets L/R ” button is used for calculating the left and right cosets of a bomber group. The action of this button on the jumping group on 5 letters is shown in Figure 10

Figure 10: Calculation of the left and right cosets of a subgroup

The “ centralizer ” button is used for calculating the centralizer of selected elements. When multiple elements are selected, the centralizer of each component is computed. The action of this button is shown in Figure 11.

Figure 11: Calculation of the centralizer of group elements

The “ span ” button is used for measuring a group that is generated by the selected component ( s ) . The action of this button on several selected group elements is shown in Figure12.

Figure 12: Calculation of a subgroup generated by the selected group elements

The “ group description ” button is different from the other discussed buttons, in that its calculations do non depend on the selected group elements. It gives the group ‘s name, its cardinality, conjugate categories, centre and commutator. The action of this button on the cyclic group with 17 elements is shown in Figure13.

Figure 13: The description of a selected group

Output screen

This constituent is responsible for exposing the end product of all calculations performed on the finite group analysis console. The end product screen besides interacts with the group choice and alteration console and is updated each clip the group is modified or changed. The execution of this constituent is shown in Figure2. The execution is a text country that displays the calculations of the group analysis console. It besides consists of a list that displays the group ‘s elements and a check that displays the cayley tabular array of the group. Part of the execution is shown in Figure14 while the other portion is shown in Figure 16.

Figure 14: Check that displays the cayley tabular array of the selected group

The system can be viewed logically as shown in Figure15. A function signifier the high degree position of the system to the execution is shown in Figure 16.

Figure 15: A high degree position of the Finite Group Analyzer.

Output Screen

## Finite Group choice, definition and alteration console

## Finite Group Analysis console

Figure 16: A function from the high degree position of the application to the GUI execution.

## 3.3 Discussion

## 3.3.1 Group Theory

Group theory is the survey of the formation and belongingss of mathematical groups.

## 3.3.2 Group

A group is an algebraic construction dwelling of a set and a binary operation that combines any two of its elements to organize a 3rd component that belongs to the groups set ( Fraleigh, 2003 ) . To measure up as a group, the set and the operation must fulfill a few conditions called group maxims, viz. : Closing, Associativity, Existence of a alone individuality component and the being of an Inverse component for every component in the group.

## 3.3.3 Finite Group

In mathematics and abstract algebra, a finite group is a group whose implicit in set G has finitely many elements.

## 3.3.4 Basic Group Concepts

## 3.3.4.1 Group Homomorphism

Group homomorphies are maps that preserve group construction. A map F: G a†’ H between two groups is a homomorphy if the equation

F ( g aˆ? K ) = F ( g ) aˆ? F ( K ) .

holds for all elements g, K in G

## 3.3.4.2 Subgroup

A subgroup of a group G is subset H of G such that, the individuality component of G is contained in H, and whenever h1 and h2 are in H, so so are h1 aˆ? h2 and h1a?’1.

## 3.3.4.3 Coset

A subgroup H of a group G defines left and right cosets, which can be thought of as interlingual renditions of H by arbitrary group elements g. In symbolic footings, the left and right cosets of H incorporating g are

gH = { gh |h a?? H } and Hg = { mercury | H a?? H } , severally.

## 3.3.4.4 Normal Subgroup

A subgroup N of a group G is normal in G if xN = Nx for all x in G, that is, every left coset is besides a right coset

## 3.3.4.5 Conjugate Classes Centralizers and Center of a Group

This relation defined on the implicit in set of a group G denoted as: a ~ B if and merely if b = x a x-1 where a, B, x are elements of G. in this instance a and B are said to be conjugates.

The above relation is an equality relation and is therefore a divider of G. The equality category of an component a in G is denoted as Cl ( a ) and is called the conjugate category of a.

The centralizer of an component a in a group G is defined as follows:

If a is an component of a group G, so the centralizer of a in G, denoted as CG ( a ) is given by CG ( a ) = { g Iµ G| ag = tabun }

The centre of the Group is denoted as Z ( G ) and is defined by Z ( G ) = { omega Iµ G| zg = gz for all g Iµ G } .

## 3.3.4.6 Commutator Subgroup

The commutator of two elements, g and H, of a group, G, is the component

[ g, H ] = ga?’1ha?’1ghIt is equal to the group ‘s individuality if and merely if g and h commute ( i.e. , if and merely if gh = mercury ) . The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G.

## 3.3.5 Important Classs of Groups Studied At Under Graduate Course in Abstract Algebra

3.3.5.1 The Symmetric Group on n Letters ( Sn )

The symmetric group on a set Ten is the group whose implicit in set is the aggregation of all bijections from Ten to X and whose group operation is that of map composing. The symmetric group of grade N is the symmetric group on the set X = { 1, 2, … , n The symmetric group on a set of N has order N! and is abelian if and merely if n a‰¤ 2.

## a. Elementss and Operation on the Symmetric Group

The binary operation in the Symmetric group is composing of maps. Let

The above shows two ways of denoting a substitution the first signifier is called rhythm notation while the 2nd is called array signifier. The two signifiers are tantamount.

The binary operation defined on the symmetric group is composing of maps. See the maps degree Fahrenheit and g

Then the map degree Fahrenheit I? g ( normally denoted as f g ) is the composing of degree Fahrenheit and g.

## b. Transpositions and Parity of Permutations

In the cyclic notation of a substitution whose rhythm notation consists of a individual rhythm with two elements is called a heterotaxy. Every substitution can be expressed as a merchandise of heterotaxies. Therefore any substitution of a finite set of at least two elements is a merchandise of heterotaxies. When a substitution is expressed as the merchandise of heterotaxies, the figure of heterotaxies is either ever even or odd. If the figure heterotaxies are even so the substitution is said to be even otherwise it is even ( hypertext transfer protocol: //www.en.wikipedia.org/wiki/Symmetric group ) .

3.3.5.2 The Alternating Group on n Letters ( An )

The jumping group is the group of even substitutions of a finite set. The jumping group on the set { 1, … , n } is called the jumping group of grade N, or the jumping group on n letters and denoted by An

## 3.3.5.3 Dihedral Group ( D2n )

The dihedral group is the group of symmetricalnesss of a regular polygon, including both rotary motions and contemplations. Dihedral groups are among the simplest illustrations of finite groups, and they play an of import function in group theory, geometry, and chemical science.

Figure 17 shows the consequence of composing in the group D3 ( the symmetricalnesss of an equilateral trigon ) . R0 denotes the individuality ; R1 and R2 denote counterclockwise rotary motions by 120 and 240 grades ; and S0, S1, and S2 denote contemplations across the three lines shown in Figure 2.2.

Figure 17: Rigid gestures on an equilateral trigon

For illustration, S2S1 = R1 because the contemplation S1 followed by the contemplation S2 consequences in a 120-

In general, the group Dn has elements R0, … , Rna?’1 and S0, … , Sna?’1, with composing given by the undermentioned expression:

In all instances, add-on and minus of inferiors should be performed utilizing modular arithmetic with modulus Ns ( hypertext transfer protocol: //www.en.wikipedia.org/wiki/Dihedral group ) .

## 3.3.5.4 Cyclic Group

In group theory, a cyclic group is a group that can be generated by a individual component, in the sense that the group has an component g ( called a “ generator ” of the group ) such that, when written multiplicatively, every component of the group is a power of g.

A group G is called cyclic if there exists an component g in G such that

G = & lt ; g & gt ; = { gn | N is an whole number } .

## 3.4 The Finite Group Analyzer

Finite groups form a really particular category of groups in that the construction of many categories of finite groups have been to the full studied both analytically and by the usage of computing machines to detect their assorted constructions. Computer Programs such as Group Analyzer version 7, developed in Catholic University of America for Macintosh computing machines running on MacOS 8.1 through 9.x ( hypertext transfer protocol: //math.cua.edu/glenn/ga7page.htm ) and Java codification developed at San Diego State University by Jeffrey Barr, for the coevals, designation and analysis of finite groups ( hypertext transfer protocol: //www.groovypower.com/thesis ) , can easy cipher most of the cardinal group constructions.

The Finite Group Analyzer is designed to let users experiment with the construction and representation of the group in simple and intuitive mode, it besides allows the users to acquire speedy replies to “ what if ” inquiries refering the construction and representation of finite groups. The Finite Group Analyzer does non detect groups, but provides users with a simple interface to stipulate the group they wish to work with. Once a group has been specified the user can so calculate all the constructions supported by the application with regard to the specified group. The application allows the user compute finite group constructions, which are impractical to calculate manually. A cardinal characteristic of the Finite Group Analyzer is that it allows the user to stipulate the twine representation of the group ‘s elements. This characteristic is of import because a group may hold several widely used representations of its elements. Another characteristic of the Finite Group Analyzer is that elements of a group have a lexicographic ordination, which is ever used when elements of a group are displayed. This allows elements of a larger group to be easy identified and worked with.

## 4.1 Decision

The developed application ( Finite Group Analyzer ) provides flexible and intuitive graphical user interface ( GUI ) for analysing finite groups. The Finite Group Analyzer provides a broad assortment of finite groups for users to analyse and can therefore assist abstract algebra pupils to acquire an innate apprehension of the construct of groups, which forms and built-in portion in the survey of abstract algebra.

The Finite Group Analyzer removes to a great extent the obstruction created by the figure of elements in a group, since groups with more than seven elements can take rather a important sum of clip to analyse without utilizing analytical methods.

## 5.1 Recommendation

The Finite Group Analyzer was designed in a manner that makes it easy to widen categories of groups supported by the system. Therefore, it recommend that categories of finite Groups that are studied by pupils but are non available in the Finite Group Analyzer be implemented and added to the system.

Algorithms exists for the rating of more advanced finite group constructions such as direct merchandises of groups and coevals of groups based on relationships between generators of the group. It is recommended that the system be extended to back up more advanced group analysis such as calculation of the direct merchandise of groups, specification of group homomorphies. The betterments must be supported by improved user interface for analysis.