Project Materials

Determination of the Number Of Non-Abelian Isomorphic Types of Certain Finite Groups

ABSTRACT

The first part of this work established, with examples, the fact that there are more than one non-abelian isomorphic types of groups of order n = sp, (s,p) = 1, where s

1 was worked out and such groups have no non-abelian isomorphic types. This gave credence to the fact that a group of order 15 and its like do not have a non-abelian isomorphic type.

It also generated the non-abelian isomorphic types of groups of order n = spq, where s, p and q are distinct primes considering the congruence relationships between the primes. It was seen that there are more non-abelian isomorphic types when q 1 (mod p), q  1 (mod s) and p  1 (mod s).

When q is not congruent to 1 modulo p but congruent to 1 modulo s fewer non-abelian isomorphic types were obtained. Moreover, if q is not congruent to 1 modulo p, q not congruent to 1 modulo s, and p not congruent to 1 modulo s, there cannot be a non-abelian isomorphic type of a group of order n = spq.

In this case groups of order n = 2pq, 3pq, 5pq and 7pq were considered. Later, proofs of the number of non-abelian isomorphic types for n =sp and n =spq using the examples earlier generated were given.

CHAPTER ONE

INTRODUCTION

1.0 BACKGROUND OF STUDY

Group Theory is relevant to every branch of Mathematics where symmetry is studied. Every symmetrical object is associated with a group. It is due to this association that groups arise in different areas like Quantum Mechanics, Crystallography, Biology, and even Computer Science. There is no such easy definition of symmetry among objects without leading its way to the theory of groups.

Classifying groups arise when trying to distinguish the number of isomorphic groups of order n. In organic chemistry, conformal factors affect the structure of a molecule and its physical, chemical and biological properties. For instance, the position of atoms, relative to one another affects the structural formula of Hydrogen peroxide, H2O2.

We could write two different planar geometries that differ by a 1800 rotation about 0 – 0 bond. According to Francis A Carey (2003) one could also write an infinite number of non planar structures by tiny increments of rotation about the 0 – 0 bonds; Francis A Carey (2003).

Groups may be presented in several ways like multiplication table, by its generators and relations, by Cayley graph, as a group of transformations (usually a geometric object), as a subgroup of a permutation group, or a subgroup of a matrix group to mention a few.

1.1 STATEMENT OF THE PROBLEM

Classifying groups arise when trying to distinguish the number of isomorphic types of a group of order n. Hall Jnr and Senior (1964) used invariants as the number of elements of each order k (k small) to determine whether two groups of order 2n (n < 6) are isomorphic. Philip (1988) in his article developed a systematic classification theory for groups of prime power orders.

For certain classes of groups, they exists practical methods to list such groups. Newman and O’Brien (1990) introduced an algorithm to determine up to isomorphism the groups of prime-power order. The determination of all groups of a given order up to isomorphism is an old question in group theory. It was introduced by Cayley who constructed the groups of order 4 and 6 in 1854.

MeubÜser (1967) listed all groups of order at most 100 except for 64 and 96. The groups of order 96 were added by Lane (1982).

Moreover, for factorizations of certain orders, the corresponding groups have been classified, e.g. Holder (1983) determined the groups of order pq2 and pqr, and James (1980) determined the groups of order pn for odd primes and n < 6.

Recently, algorithms have been used to determine certain groups. For example O’Brian (1991) determined the 2-groups of order at most 28 and the 3-groups of order at most 36. Moreover, Betten (1996) developed a method to construct finite soluble groups and used his construction to construct soluble groups of order at most 242.

Determination of isomorphic types has been a comparatively difficult problem as there was no method that is sufficiently effective.

Most of the classifications of the non-abelian isomorphic types of certain finite groups were done for groups of small orders. This is possibly due to the complexity of computation as the factors increase. The problem then arise to find the non-abelian isomorphic types of groups of higher orders which can be factorized into two or three distinct primes taking into consideration of the relationship between the prime factors. The need also arise to construct a suitable computer program to assist in solving such a problem.

Hence, the statement of the problem is “Determination of the Number of non-Abelian Isomorphic Types of Certain Finite Groups”.

1.2 AIM AND OBJECTIVES

The aim of this thesis is to determine the number of non-abelian isomorphic types of certain finite groups of higher orders.

We hope to achieve the following objectives:

(i) Finding relationship, through series of examples, of the number of non-Abelian Isomorphic types of groups of order n=sp and the congruence relation between the primes s and p.

(ii) Determining the proof for the number of non-Abelian isomorphic types in each congruence relationship and stating their defining relations.

(iii) Determine and design a suitable computer program that will help in working out the number relationship between such primes and generating the numbers for the non-Abelian isomorphic types.

(iv) Finding the non-Abelian isomorphic types of groups of order n = spq where s,p and q are distinct primes and determining their defining relations.

1.3 SCOPE OF THE STUDY

The scope here is limited to the determination of the number of non-Abelian isomorphic types of groups of order 2p, 3p, 5p, 7p, 11p, 13p where p < 4000. Also considered are groups of order 2pq, 3pq, 5pq and 7pq. The primes p and q are distinct primes with p < q.

1.4 DEFINITION OF THE CONCEPT OF ISOMORPHIC GROUPS

The concept of group isomorphism can be explained with chessboard that has four plane symmetries. The identity, rotation r through  about its centre, and the reflections 1 2 q , q in its two diagonals form a group under composition whose multiplication is given in table 1 below.

Table 1.1: Four plane symmetries of a chessboard
 e r 1 q 2 q
e e r 1 q 2 q
r r e 2 q 1 q
1 q 1 q 2 q
e r
2 q 2 q 1 q r e

It is easy to check that multiplication modulo eight makes the numbers 1,3,5,7 into a group.

There is an apparent similarity between these two groups if we ignore their origins. In each case the group has four elements, and these elements appear to combine in the same manner. Only the way in which the elements are labeled distinguishes one table from the other.

Label the first group G, the second G’, and the correspondence.

1, 3, 5, 7 1 2 e r q  q  ,

This correspondence is called an isomorphism between G and G’. It is a bijection and it carries the multiplication of G to that of G’. Technically they are isomorphic in the following sense.

Two groups G and G’ are isomorphic if there is a bijection  from G to G’ which satisfies xyxy for all x, yG. The function  is called an isomorphism between G and G’.

Hence the isomorphism as a bijection implies that G and G’ must have the same order.

It sends the identity of G to that of G’. Isomorphism also preserves the order of each element (Armstrong, 1988).

1.4.1 EXAMPLES

(i) The group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication ( + × ) R , .

Proof: Define f :(R,+) →( + ×) R , by f(x) = ex. For elements x, y in R then
f(x) = f(y) then ex = ey, so x = y. This implies that x ≠ y, then f(x) ≠ f(y) i.e., ex ≠ ey.
If r is an element o R+, then f(ln r) = r, where ln r belong to R showing that f is onto
R+ . Again, for elements x, y in R, we have
f(x + y) = ex+y = ex.ey = f(x)f(y).
Hence (R,+) is isomorphic to( + ×) R , .

(ii) Every cyclic group of infinite order is isomorphic to the additive group I of integers

Proof: Consider the infinite cyclic group G generated by a and the mapping
n → an, n∈ I of I into G.

Now, this mapping is onto since any n in I is mapped to exactly one an.

Moreover, it is one-to one since if s > t we have s ↔ as and t ↔ at, then as-t =1 and G would be finite. Hence if s ‡ t, then as ≠ at.

Finally, s + t ↔ as+t = as.at. Hence the mapping is an isomorphism, that is I  G.

(iii) The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S’ of complex numbers of absolute value 1 (with multiplication):

R/Z  S’

An isomorphism is given by
f(x+Z) = e2xi
for every x in R.

Proof: We only need to show that for any k ε Z, then f(x + k) = e2i(x +k) = e2xi +2iπk = e2xi. e2ki = e2xi(Cos2πk + iSin2πk) = e2xi.

If x ≠ y then f(x + k) ≠ f(y + k), i.e., e2xi ≠ e2yi . Also for z ε R, then f(ln(z + k)) = eln(z + k) = z + k.

(iv) The Klein four-group is isomorphic to the direct product of two copies of Z2 = Z/2Z and can therefore be written Z2xZ2. Another notation is D2, because it is a dihedral group.

(v) Generalizing this, for all odd n, D2n is isomorphic with the direct product of Dn and Z2.

1.4.2 PROPERTIES OF ISOMORPHIC GROUPS

(i) The Kernel of an isomorphism from (G,*) to (H, ʘ), is always {eG} where eG is the identity of the group (G,*).

(ii) If (G,*) is isomorphic to (H,ʘ), and if G is Abelian then so is H.

(iii) If (G,*) is a group that is isomorphic to (H,ʘ) [where f is the isomorphism], then if a belongs to G and has order n, then so does f(a)

(iv) If (G,*) is a locally finite group that is isomorphic to (H,ʘ), then (H,ʘ), is also locally finite.

We state mostly without proof certain fundamental results of group theory which we shall be needed:

1.4.3 THEOREM (LAGRANGE’S THEOREM)

Let G be a group of finite order n, and H a subgroup of G. The order of H divides the order of G.

1.4.4 THEOREM (CAUCHY’S THEOREM)

If p is a prime number and p G then G has an element of order p.

1.4.5 THEOREM (SYLOW’S FIRST THEOREM)

If pa is the highest power of a prime dividing the order of a group G, then G has at least one subgroup of order p

1.4.6 DEFINITION 1

For any prime, p, we say that a group G is a p-group if every element x ,iny GG has order pk, for some integer k

1.4.7 DEFINITION 2

Let G be a finite group of order n = pq, where (p,q) = 1. Then any subgroup of order pm is called a Sylow p-subgroup of G.

1.4.8. DEFINITION 3

Let a be an element of a group G and e the identity element of G. The smallest positive integer n such that an = e is called the order of a. The order of a group G, written G is the cardinal number of elements of G. G is said to be finite or infinite according as its order is finite or infinite (Kuku, 1980).