Classifying Groups of Small Order

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.


Introduction
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 2 3 ×   and 2 2 ×   , the former is cyclic whereas the latter is not.Here p  denotes the set of integers 0,1, 2, , 1 p −  that are added modulo p so that it is perhaps better to write 2 3 ⊕   rather than 2 3 ×   .Many authors use the notation p C to connote a cyclic group of order p to which of course p  is isomorphic.In particular in the sequel it is sometimes preferable to think of 2  as being isomorphic to the group consisting of the integers { } 1 ± under multiplication.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 [1].Another pillar in the classification of finite abelian groups is Sylow's Theorem or Theorems depending on how they are presented [2].Important as these Theorems are, the proofs probably go over the heads of first semester abstract algebra students even though the statements of the theorems are not difficult and numerical examples come readily to hand.In Gallian's book [3] Sylow's Theorems are relegated to a section titled "Special Topics" and appear on page 399.We recommend [3] for all details not covered adequately in this short article.The author still enjoys the book by Herstein [4] from which he originally learned the subject.Another very nice book, among many others, is written by Rotman [5].So that is the program: classify groups of order less than 16 knowing the structure of finite abelian groups without using Sylow theory.
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 n D of order n.In Section 6 we prove the main Theorem that identifies n D in various cases.In Section 7 we take stock and see which values of G have already been accounted for and then settle the remaining 4 cases.In Section 8 we draw a few conclusions and finally in Section 9 we present a few exercises for students.
We shall say a few words about notation: The identity element in G will be denoted by e.We shall denote the center of the group Z G and the centralizer of an element g G ∈ , by We shall use the word order frequently with different meanings.Thus G denotes the order of G, that is, the number of elements in G.We shall also use the same notation if merely Y G ⊂ .On the other hand, ( ) o g is the order of an element g G ∈ , that is to say, the smallest positive power is r such that r g e = .The reason for using the same word is that ( ) o g is also the order of the cyclic subgroup generated by g and that cyclic subgroup will be denoted by g .Another not very serious remark: I have consistently used numbers rather than writing numbers in words because there are a lot of them and I considered that to do so had a much of a more visual impact.

Some History
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 [6] which can serve effectively as a text for a more advanced course on group theory.The theory was a gradual coalescence of ideas distilled from the areas of polynomial equations, number theory and geometry.Indeed the very concept of "geometry" itself was being expanded by Gauss, Riemann, Lobachevsky and Bolyai in the first part of the nineteenth century.Klein's Erlangen Program, which associates a group of symmetries to a particular flavor of geometry, dates from 1872.There had been numerous contributions to the theory of groups made by the likes of Lagrange and Cauchy (whose theorems in finite group theory we still celebrate today), Euler and Gauss himself, particularly in what we refer to nowadays as abelian groups and their relationship to modular arithmetic.Apparently Galois in 1831 had begun to grasp the notion of an abstract group, as opposed to a group of permutations, as too did Cayley in the 1850's although both men's work was years ahead of its time.
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 [7] was published as long ago as 1897.These books were enormously influential.Meanwhile the Norwegian Ludwig Sylow's fundamental set of Theorems had appeared as early as 1872 [2]; it is interesting that his paper was written in French whereas Sophus Lie, Sylow's equally famous compatriot, wrote in German.Notice that Sylow refers again to "les groupes de substitutions", or permutation group in more modern language.Burnside [7] credits Sylow with laying the first real theoretical foundations of group theory.By 1870 Jordan had proved the Jordan-Hölder theorem for permutation groups and Hölder proved it for abstract groups in general in 1889.Furthermore, Hölder in 1893 was studying groups of order 3 2 , , p pq pqr and 4  p .In America in 1900 G. A. Miller and G. H. Ling [8] proved that there was no simple group of order between 1092 and 2001.In the negative direction, as late as 1908 Burnside was complaining to the London Mathematical Society about the lack of acceptance in Britain of group theory as a subject in its own right.
By 1930 Miller [9] was announcing the classification of all groups of order up 100.The most difficult case, as Miller's paper suggests, was 64 G = .Apparently Miller's paper contained errors that were not corrected until 1964 by Marshall Hall and J K Senior [10].From the 1960's onwards the quest for the classification of the finite simple groups was in full swing.Copious details may be found on the Wolfram website that do not need to be repeated here.We will be content simply to cite [11] [12] as more traditional references.Apparently the complete classification was not achieved in toto until 2004.

Elementary Facts
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 ( ) Z G is its center then the factor group ( ) for some finite p) and H and K are complementary in the sense that , then G H K ≈ × , the direct product.

Group Actions
Let G be a group and X a set.Then we say G acts on X if there is a group homomorphism φ from G to X S , the latter set denoting the group of permutations of X.Here are three examples of group actions where the group acts on itself, that is, Orb x of x and define a relation on X according as elements belong to the same orbit.It is left as exercise to show that this relation is actually an equivalence relation.Define also ( )

Stab x or x
G to be the stabilizer subgroup of x, that is, In the case of the first two actions defined above there is only one orbit for each g G ∈ , that is G itself; in that case we say that the action is transitive.Also, for each g G ∈ we have However, in the third example of the action of conjugation, the orbits are precisely the conjugacy classes of G and g G is the centralizer It is easy to show that g G G < .Notice also that conjugation does nothing at all for abelian groups!In fact conjugation is a very special kind of action because G acts on itself not merely as a set but as a group; in other word the action maps not only to G S but to

( )
Aut G , the group of automorphisms of G.
Theorem 1.There is a one to one correspondence between elements of ( ) ; then there exists g G ∈ such that y gx = .We obtain a map from ( )

Orb x to
x G G by mapping g to the left coset of x gG in G.We leave the student to verify that this map is well-defined (the hardest part for students) and bijective.

Example
The group D 6 or 3 , , , , , e x x y xy x y subject to the relations 3 2 2 , x y e yx x y = = = .The last relation used repeatedly enables us to "twist" products of x and y to work all the x's to the front in a string of x's and y's and all the y's to the back.Then the other relations may be used to reduce any string to one of the 6 elements that constitute the group.The conjugacy classes of 6 , , , , , e x x y xy x y so that x G e x x = and 1 2 , , , ,

Conjugation and Centralizers
The class equation expresses G as a sum of the orders of the conjugacy classes.
provided G is not the trivial group consisting of a single element.Hence on the right hand side of the class equation we will have as many 1's as appear in ( ) Z G and sums of various proper divisors of G .

The Dihedral Group
The group 2n D of order 2n is generated by x and y subject to the relations , , x is a singleton and ~ signifies "is conjugate to".Now

Theorem
Lemma 1.If G is a non-abelian group of order pq where p and q are distinct primes then Z G divides G by Lagrange's Theorem and hence

( )
Z G can only be one of 1, , , p q pq .Now ( ) is cyclic of order p.We cannot have ( )

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

( )
Z G G ≠ .Since groups of prime order are cyclic we do not need to consider the cases 2, 3, 5, 7,11,13 G = .Furthermore Theorem 1 disposes of the cases 6,10,14 n = so it remains to study the cases 8, . Hence we can only have that

15 G =
By Lemma 1 and the class equation gives 15 1 3 3 3 5 = + + + + .The elements of the conjugacy class of order 5 are themselves of order 3: they cannot be of order 5 because their centralizers are of order 3 whereas an element of order 5 has a centralizer of order at least 5. On the other hand each of the elements in the 3 conjugacy classes of order 3 must have order 5: indeed, if x is such an element then x .Now the total number of elements of order 5 must be 9, which contradicts Fact 5, since it would have to be multiple of 4. Hence there cannot exist G for which 15 G = and G is non-abelian.

G =
We know of course that 8 D is a non-abelian group of order 8.We investigate whether there are any others.According to Fact 4, G must have an element x of order 4. Suppose that y G ∈ but that y x ∉ .Then according to Theorem 2, if G is not isomorphic to 8 D we may assume both that 4 y e = and 2 2 x y = .It is then easy to see by cancelation that the elements of G must be { } Finally we can replace e by 1, 2 x by 1 − and , , x y xy by , , i j k , respectively, so as to obtain the quaternion group Q usually written as { } 1, , , ± ± ± ± i j k .

G =
We have since x y e = = and all the conditions for the existence of a group are now met.To recognize this group, introduce a new generator 2 w x = so that 3 w e = .The elements of G can be written as { } G Z Z ≈ neither of these extensions split: here 6 Z x ≈ .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 4 Z and the second the group Q studied in subsection 6.3.For further details see [13] which is another nice reference for more advanced material.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 = + + + .Each of the elements in the conjugacy classes of order 4 must have order 3 because their centralizers are of order 3. Elements in the conjugacy class of order 3 could have order 2 or 4 because each of their centralizers are of order 4; however, if the elements have order four their squares would have order 2 and there are no conjugacy classes left to accommodate them.Hence the elements in the conjugacy class of order 3 have order 2.
Let x and y be elements of order 2 and 3, respectively.Then e y x e y =   .

1 . 2 G
= and G would be abelian.Hence x belongs to a conjugacy class of 2 elements.Furthermore Suppose that G is non-abelian and that 2 y in the conjugacy class of order p. Now( )C y = and so ( )2o y = since y e ≠ .Turning to an element x in a conjugacy class of order 2 then ( ) x y y xy yx .Now yx xy ≠ or else G will be abelian.The only possibilities for ( ) o xy are 2 or 4; however, if ( ) again G would be abelian and so ( ) 4 o xy = .Again by cancelation and elimination of all other possibilities we see that

4 Z
y y w wy wy wy w w y w y w y .generated by y, acts on 3 Z generated by w via conjugation and engenders a group denoted by T that is a semi-direct product of 4

( ) 1 Z
G = .The class equation must contain another odd number which can only be 3. Hence we have the following cases:

2 y 1 Z 2 Z
index 2 it is normal in G. Hence we may write G is a positive power of x, it will be a central element.However, we are considering now only cases where ( ) G = so the only possibility is 2 G = we conclude that none of the three cases for which 2 appears in the class equation actually do occur.Now consider the last case of the class equation 12 1 3 4 4

2 yxy and 2 y
xy are of order 2 since they are conjugate to x; if any two of 2 2 , , x yxy y xy are equal we would have that xy yx = .In that case we have a cyclic subgroup of order 6 generated by xy and whose only element of order 2 is x.Now pick a second element of order 2 and appeal to Corollary 1 to conclude again that 12 G D ≈ and again contradicting ( ) 1 Z G = .Thus we may assume that 2 2 , , x yxy y xy are distinct.Now xy and yx must be of order 3.It follows that 2 xyxy y x = Thus the order of a conjugacy class is the index of the centralizer of any element x in that conjugacy class, that is, Each of the terms on the right hand side except the first must be divisors of m p since they are indexes of centralizers and cannot be 1 otherwise ( ) { } Suppose that G is a non-abelian group of order 2n where n is either an odd prime or 4 or 6.
[4]1, 2, 3, 4 or 6.We may reject the cases Z G = .Then in the class equation we must have another odd number, which can only be 3. Thus 12 1 1 1 3 = + + + + .Consider an element x in the conjugacy classes of order 3. Its centralizer is of order 4 and contains ( ) D contains a cyclic normal subgroup N of order 3.According to the correspondence theorem[4], which is really the deluxe version of the first isomorphism theorem, N engenders a normal subgroup N of order 6 in G that contains ( )