_{1}

^{*}

The classification of groups of order less than 16 is reconsidered. The goal of the paper is partly historical and partly pedagogical and aims to achieve the classification as simply as possible in a way which can be easily incorporated into a first course in abstract algebra and without appealing to the Sylow Theorems. The paper concludes with some exercises for students.

This past semester I have been teaching an introductory course on abstract algebra. The question arises of how to reach an audience of a mixed background, for example, graduate and undergraduate students. My solution was to present the material in a very computational way rather than going the usual route of lots of Theorems and Propositions. More specifically, why not orient the course towards the problem of classifying groups of small order? In the present article I shall present a classification of groups of order less than 16. The ground rules are that we shall assume that students have covered the first four weeks of group theory. In Section 3 we present six “Elementary Facts” which students can treat as homework exercises. We shall also assume that we have known the classification of finite abelian groups. Maybe that is a lot to ask, however, it is easy to understand the structure of finite abelian groups: just keep in mind the groups

Probably the nicest way to obtain the structure for finite abelian groups is as Corollary to the structure Theorem for a finitely generated module over a principal ideal domain, which can be covered nicely in the second semester of the abstract algebra class: see for example [

In Section 3 we review some historical details about the emergence of the concept of an abstract group and some of the early results on the classification of groups of small order. In Section 4 we consider the action of conjugation of a group on itself and the class equation. In Section 5 we study properties of the dihedral group

We shall say a few words about notation:

The development of group theory is a complicated historical and epistomological question that we cannot possibly do justice here. We shall not supply many of the historical references as they can be found in the excellent book by Rose [

Eventually the abstract idea of a “group” emerged. A two volume book on algebra by Heinrich Weber “Lehrbuch der Algebra” appeared in 1895 and 1896 and the first edition of William Burnside’s book [

By 1930 Miller [

1) Groups of prime order are cyclic and unique up to isomorphism.

2) Conjugate elements have the same order.

3) If G is a group and

4) If all elements of G except e are of order 2 then G is abelian.

5) If p is prime the number of elements of order p is a multiple of

6) If a group G is generated by two normal subgroups H and K (so that every

Let G be a group and X a set. Then we say G acts on X if there is a group homomorphism

・ For

・ For

・ For

Define

Theorem 1. There is a one to one correspondence between elements of

Proof. Suppose that

The group D_{6} or

The class equation expresses

class is the index of the centralizer of any element x in that conjugacy class, that is,

order of a conjugacy class divides

The group

singleton and ~ signifies “is conjugate to”.

Now

and

Assuming

whereas if n is even

A few more elementary calculations convince one that if n is odd, the class equation for

the number of 2’s being

the number of 2’s being

Notice finally that if

Lemma 1. If G is a non-abelian group of order pq where p and q are distinct primes then

Proof. Since

Lemma 2. If G is a group of order

Proof. If

Theorem 2. Suppose that G is a non-abelian group of order 2n where n is either an odd prime or 4 or 6. Suppose further that there exist

Proof. Clearly

Suppose now that n is an odd prime. Then

Suppose that

Suppose that

Hence

Corollary 1. Suppose that G is non-abelian and that

Proof. By 1

Turning to an element x in a conjugacy class of order 2 then

For each order bigger than 5 we are looking for a non-abelian group G. Since G is not abelian we have that

It follows from Lemma 2 that

By Lemma 1

We know of course that

of G as

Finally we can replace e by 1,

We have since

Suppose then that

Notice that

So in order to avoid having

So

Thus the only way to avoid having

In fact the group T belongs to a familiar class of finite groups of order 4n called the dicyclic groups and also known as the binary dihedral groups. Depending how one counts, the first such group is

Now suppose that

・

・

・

・

We consider the first three cases for which 2 appears in the class equation. We take x in one of the conjugacy classes of order 2. Then

Now consider the last case of the class equation

Let x and y be elements of order 2 and 3, respectively. Then

Finally, a more sophisticated approach is to note that the conjugacy class of order 3 together with e forms a normal subgroup

We have classified all groups of order < 16 without using Sylow theory and assuming we have known the classification of finite abelian groups. It seems remarkable to the author that for

In [

・ Supply proofs of the six Elementary Facts. As a hint for the sixth, note that it suffices to map generators to generators.

・ Show that the orbits

・ Show that for the stabilizer subgroup of

・ Finish the details of Theorem 4.1.

・ Find 8 mutually non-isomorphic groups of order 16.

・ Find generators and relations for the group

・ Find an explicit isomorphism between

The author thanks Paul Hewitt for stimulating discussions and some valuable suggestions from the referees for improving many of the arguments.

GerardThompson, (2016) Classifying Groups of Small Order. Advances in Pure Mathematics,06,58-65. doi: 10.4236/apm.2016.62007