Group Theory
Created | Updated Oct 2, 2006
Group Theory is the study of objects in mathematics called groups. The concept of the group crops up in many areas of mathematics and science, and so this entry aims to provide a guide to what groups are, along with some of their properties and results that relate to them.
What is a Group?
A group consists of two parts; a set and a mathematical operation. The set is essentially a collection of elements, such as numbers, that are part of the group. The operation1 must be a binary operation, meaning that it should take two elements as inputs and output another element. For example, we can regard multiplication as a binary operation, as we can multiply two elements to get an output of another element. The presence of an operator therefore allows us to produce another element from two of the elements in the group, thereby linking different elements together.
We denote the operation using a symbol, common examples of which are the '+' used in addition and the '×' used in multiplication. When describing rules that apply to all binary operations, we will use the generic symbol '*' (a subscript asterisk) to denote an operation. We denote the result of applying an operation '*', to elements a and b, to be a*b. For example, if we add a to b we get a+b as a result.
If we denote our set as G, and the operation *, then the group can be written in shorthand as (G,*). In this Entry we will simply call the group G, unless it is necessary to clarify the binary operation we are referring to. Also, it will be convenient to write a*b as ab, as these two expressions have the same meaning for any particular operation.
Conditions
In fact, not every binary operation will turn a set into a group. There are certain conditions that must be met, to turn a set G and operation *, into a group (G,*). These are as follows:
1 - Closure
The first condition is that if a and b are in our set G, then a*b is also in G. This simply means that our set is self-contained, and we do not need to consider anything outside the set G when applying our operation.
2 - Identity
This condition states that there is at least one element e of G, such that x*e = e*x = x, for any x in G. Such an element is called an identity element of G, and doesn't affect itself or other members of the group.
For example, if G is the set of whole numbers, and * is the addition (+) operation, then 0 is an identity, as adding 0 to any number doesn't change it (x+ 0 = 0+x = x). Also, if G is the set of rational numbers2 except for 0, and * is multiplication (×), then 1 is an identity (x ×1 = 1×x = x). Sometimes 0 and 1 are used to denote identities in other situations where operations are defined that share properties of addition and multiplication respectively.
Note that we haven't said that only one identity element exists for any operation, although this is true, as we shall see later.
3 - Inverse
This condition states that for each x in G, the group must contain at least one element y, such that x*y = y*x = e. This makes y the 'inverse' of x, as it cancels out x. For example, if G is the set of whole numbers, and * is addition, then the inverse of x is -x. Each element has only one inverse, as we shall see later, and we write the inverse of x as x-1.
Note that if y is the inverse of x, then x is the inverse of y. We can pair off all members of G with their inverses, with some exceptions. Some elements of G will be self-inverses, meaning that they are their own inverses. The identity element must be its own inverse, for example.
4 - Associativity
This condition states that x*(y*z) = (x*y)*z for any choice of x, y and z in G. The brackets here indicate that we evaluate the operation in the bracket first. For example, to perform 3 + (4 + 5), we add the 4 and 5 to get 9, leaving 3 + 9 = 12. This condition allows us to ignore brackets when evaluating an expression, as all that matters is the order the elements are written in. In the expression 3 + 4 + 5, for example, it does not matter whether we add 3 to 4 to get 7, and then add 5, or add 4 to 5 to get 9, then add the 3. Note that this doesn't necessarily mean that the order in which the elements are written doesn't matter.
Abelian Groups
If x*y = y*x, for any x and y in G, then the operation * is commutative. If G is group, then if its operation * is commutative, it is an Abelian group, or just Abelian3. Many common operations in mathematics are commutative. For example, addition and multiplication are commutative, on the set of whole numbers. However, there are some non-commutative operations, such as subtraction. For example, 5 - 2 = 3, whereas 2 - 5 = -3. The operations of subtraction and division are neither associative nor commutative, and therefore cannot be used as the main operations in groups.
The Result
These laws give us a set of elements which includes an identity, an inverse of every element in the group, and every element that can possibly be generated from these elements using the operation. The main operation used must be associative, and so the same result is achieved whenever the operation is applied to three elements written in a particular order.
Some Inherent Properties
From these initial conditions, we can prove fairly simply some elementary properties of groups.
First we will show that there is only one identity element. Suppose that there are two identity elements e and f. Then e*f = e, as f is an identity element. But e*f = f, as e is an identity element, thus e = f.
Secondly, we can show that inverses are unique. Consider an element x, and suppose we have two inverses y and z. x*y = x*z = e, as both y and z are inverse of x. But then x*y = x*z, and by multiplying by y on the left we get y*(x*y) = y*(x*z), and by associativity (y*x)*y = (y*x)*z. Finally, y*x = e, thus y = z.
We can also show that if k*a = k*b or a*k = b*k, then a = b. If k*a = k*b, then multiply both sides by k-1 to get k-1k*a = k-1k*b. But k cancels with k-1, so a = b. A very similar argument tells us that a*k = b*k implies that a = b. These results are called cancellation laws.
Cayley Tables
A Cayley table4 is a way of summarising the properties of a finite group5.
The Cayley table for a group consists of a table with n + 1 rows and columns, where n is the number of elements in the group. The top left box will contain the symbol for the operation defined on the group, eg, *. The other boxes in the top row contain the elements of the group, and similarly the left-hand column, both in the same order.
The other boxes contain the result of applying the operation with the element in the left-hand column in the same row on the left, and the element in the top row in the same column on the right. This is equivalent to multiplication tables, where the operation is multiplication. However, the order in which we perform the operation may matter, if the group is non-Abelian.
Below is an example of a Cayley table, for a group K, with 3 elements:
| +3 | 0 | 1 | 2 |
| 0 | 0 | 1 | 2 |
| 1 | 1 | 2 | 0 |
| 2 | 2 | 0 | 1 |
This is the Cayley table for the group consisting of the set {0,1,2}6, with the operation +3. This operation is called 'addition modulo 3'. It is similar to ordinary addition, but after addition, both numbers are replaced by their remainders when divided by 3. For example, 1+32. 1 + 2 = 3, but 3 divides by 3, so has remainder 0, thus 1+32 = 0. This is also called clock arithmetic, as it resembles the way that two hours after 11 o'clock is 1 o'clock, the result of adding 11 and 2 modulo 12.
The Cayley table has various properties, corresponding to the definition of the group, and some other properties of groups:
We can see the closure property, as the only elements present in the table are those listed on the top row and left-hand column.
We can see that 0 is an identity element, as the 0 row and 0 column are identical to the top row and left-hand column respectively. This shows that it doesn't affect any of the elements of the group.
We can observe that two elements x and y are inverses of each other, by the fact that the element in the x row and y column (or vice versa), is the identity. For example, we can see that 1 and 2 are inverses of each other, as the element in the 1 row, and 2 column (and the 1 column and 2 row), is 0, the identity.
The table, apart from the left column and top row, is a Latin square. That is, it contains each element once on each row, and once on each column. All groups will give this property. This is due to the cancellation laws mentioned, and the closure property. One of the laws states that k*a = k*b only if a = b. In other words, if a and b are different elements, then ka and kb are different. On each row we have ka, kb, kc.., where a, b, c are different, so k*a, k*b, k*c, the elements of the boxes, will be different. Each of a, b, c must appear due to the closure property. Thus each element must appear once, and only once, in each row. Similarly, the other cancellation law means that each element must occur once and only once in each column.
The fact that this group is Abelian can be seen as the group is symmetric about a line from top-left to bottom-right. A reflection through such a line would swap the rows and columns. The only distinction between rows and columns is the order that the operation is carried out in, and order doesn't matter for Abelian groups.
Unfortunately, the Associativity property cannot be verified merely by looking at the Cayley table, as the Cayley table only looks at pairs of elements, and the associativity property concerns trios of elements.
Cyclic subgroups
A subgroup H, of a group G, is a group where all of its elements are elements of G. A cyclic subgroup is a particular type of subgroup, being the smallest one containing a particular element p, with all the other elements in the subgroup being generated by p. This is written as <p>, and called the subgroup of G generated by p.
Consider a group G, and one element, p. We wish to find <p>, the smallest subgroup that contains p. Thus, we want <p> to contain only those elements of G that it must to be a group.
We start with the identity element e, and p. But if p is in <p> then p*p must be, by the closure property. Similarly p*p*p = (p*p)*p must be in <p>. This can continue on to give us that p*p*p*p, p*p*p*p*p, etc, must all be in <p>. We can write n lots of p as pn7, so p3 = p*p*p, and we can define p0 = e
Then, we need each pn to have an inverse p-n. This will constitute n lots of p-1, the inverse of p, as n lots of p will cancel with n lots of the inverse, to get e.
So, we have established that our group <p> must contain pn for all whole numbers n (positive, negative or zero). We can also see that the set <p> only contains these elements, as the set H, containing these elements, fulfils all the rules for being a group:
<p> is closed, as pn*pm = pn+m, which is also a member of H8.
It includes the identity element e = p0.
Each element pn has an inverse p-n.
Finally the associativity property applies, as it does in our larger set G, so the set H is a group, and thus <p> = H
We can also see that <p> will be Abelian, as pn*pm = pm*pn = pn+m.
If <p> is the whole of the group G, then G itself is called a cyclic group. It is clear that a cyclic subgroup will be a cyclic group by this definition, as it is the cyclic subgroup of itself generated by our original p. By this definition, we don't need to consider a cyclic group as being part of some larger group, we can check that a group is cyclic by checking that one element of the group generates the whole group.
There are essentially three types of cyclic group. The first simply consists of only an identity element. It can only be generated by an identity element, as any other element will generate a group containing itself and the identity, so must contain at least two elements.
The second consists of a single element p, and its 'powers', p2..,pn = e, where n is some positive integer and e is the identity. Here the other 'powers', such as pn+1, or p-3, are equal to one of the pi with i between 1 and n. In other words, after a certain stage, no new elements are being produced and the group is complete.
For an example of this, consider the group K considered earlier, and a particular element 1. The group <1> here will consist of 0, 1, 1+31 = 2,1+31+31 = 2+31 = 0, etc, and their inverses. But here, we cover all the elements of the group K in the subgroup. This means that K itself is a cyclic group. In fact, if we consider the group G consisting of the set {0, ..., n-1}, and the operation of addition modulo n, then G is a finite cyclic group, and in fact all cyclic groups are in some way equivalent to one such group. We call such groups Zn.
We can observe that the subsequent rows of its Cayley table are copies of the row above, shifted one place to the right. This is true for all finite Cyclic groups.
The final category is infinite cyclic groups. Here we need all the 'powers' listed, as they all produce different elements. Consider for example (Z,+), the group consisting of the set of whole numbers under the operation of addition. <1>. The element 1 generates the whole of (Z,+) as any whole number n can be written as n lots of 1 added together, or -n lots of -1 added together, or 0. Thus (Z,+) = <1>, and so it is cyclic. In a sense, which we will explore later, all infinite cyclic groups are essentially the same as (Z,+).
Lagrange's Theorem
Lagrange's Theorem is one of the first 'big' results that most mathematicians see in group theory. It is named after Italian mathematician and astronomer Joseph Louis Lagrange (1736 - 1813), responsible for a number of contributions to physics and mathematics.
One of the consequences of the theorem is that the order of (number of elements in) a group G is cleanly divisible by the order of any subgroup H. This result allows us to prove a result about cyclic groups, that in fact all groups with prime numbers as orders are cyclic.
To show this, consider a group G with n elements, where n is a prime number. Then consider an element p other than the identity, and consider the cyclic subgroup H generated by it. This is a subgroup, so its order must be either 1 or n, as those are the only numbers that divide n. But H cannot be order 1, as it contains both the identity and p, thus it has at least 2 elements. So H must have order n, the same as G, thus as H is contained within G, H must be G. Thus G has a cyclic subgroup H that contains all of G, and so G is cyclic.
Isomorphism
Isomorphism is a concept that formalises what it means for two groups to be essentially the same. It is an important concept not only in group theory, but in many other areas of mathematics. First we will see a sketch of the general concept of isomorphism, then a more concrete explanation.
To get the general idea of what isomorphisms are, consider stories, and what it means for two stories to be essentially the same. One idea, which we will consider here, is that two stories are the same if the relationships between the characters in both stories are the same. To show this, we would need to pair off the characters in the stories, so that the relationship between each pair of characters in one story is the same as that between the corresponding characters in the other story.
For example, if we have two love stories, one where Alice and Barry plan to marry, but Barry meets another woman Carol, and a second where Amelia and Bob's planned marriage is threatened by Bob's meeting of Cathy. Here we can pair off Albert with Arthur, Barry with Bob, and Carol and Cathy. Then the relationship between Alice and Barry (planned marriage) is the same as that between Amelia and Bob, and relationships hold for each other pair of characters. Thus we can say, according to this idea, that the two stories are essentially the same. You may wish to consider more details of each story, such as the ages of the people involved, the setting, etc, but most comparisons will involve pairing the elements of one story with those of the other. If we have such a pairing off of elements, then we can say informally that the two stories are isomorphic, or that there is an isomorphism between them. In particular we can call such a pairing an isomorphism.
In groups, the idea of isomorphism is very similar, although more precise. Suppose we have two groups G and H with associated operations *1 and *2 respectively. Then a pairing that associates x in G with x' in H is an isomorphism, if a*1b is associated with a'*2b', for any a and b in G. G and H are isomorphic if some isomorphism exists between them.
If two groups G and H are isomorphic, then their Cayley table will be equivalent, in the sense that if the elements of G are replaced with the associated elements of H, then its Cayley tables will be identical to that of H. Given that the Cayley table gives all the information about the structure of a group, then this means that two isomorphic groups are essentially the same group with different labels for their elements.
Implications of Isomorphism
Two Isomorphic groups have essentially the same structure, with the only difference being how they are labelled. This means that they share the same properties. For example, if we have two Isomorphic groups, if one of them is Abelian, then so is the other. This means that if we prove one of these properties for one group, then it is proven for any other isomorphic group.
In fact, the four conditions for a set being a group are shared in a similar way. If we have two sets, each with an operation defined on them, then if one of them has (for example), the closure property, then so does the other, if they are isomorphic.
Given the idea of isomorphism, we can count the number of different groups, if we consider isomorphic groups the same. While there are infinitely many different labellings we can use there are only a finite number of groups of each order, or size, although the actual number can vary quite a lot. For example, there are only 2 groups of order 50, but 1543 of order 192.
Two further consequence refers back to the cyclic groups. Firstly it turns out that cyclic groups of the same order are isomorphic. To see this, consider two cyclic groups G and H, of order n, with operations * and + respectively. They are generated by two elements, p and q, and so consist of elements p,p*p,p*p*p,...,pn, and q,q+q,q+q+q,...,qn, respectively with the other powers being duplicates of these. Then there is an isomorphism between pm and qm. This is an isomorphism as pa*pb = pa+b, and so is associated with qa+b. But qa+b = qa+qb, so pa*pb is associated with qa+qb.
Thus all cyclic groups, of order n, are isomorphic, and isomorphic to cyclic group Zn.
This leads to the conclusion that for any prime number p, there is only one group with order p, as all the groups with order p will be cyclic, and there is essentially only one group order p, again Zp.
Further, all infinite cyclic groups are isomorphic. Again, consider two infinite cyclic groups G and H, which are generated by p and q. The groups then consist of powers of p and q respectively, and an equivalent isomorphism can be used associating pm and qm. This means that all infinite cyclic groups are isomorphic to each other. They are also therefore isomorphic to (Z,+).
