A first look at Groups

A group is a monoid in which every element has an inverse.

Fact of life.

I always chuckle when I think of that college textbook definition of a group. Innocently waggish though very correct.

While that opening definition of a group is correct, it doesn’t immediately convey the essence of a group, especially to a math noob.

Groups are ubiquitous objects in mathematics with diverse practical and theoretical applications. We are going to be looking at the meaning of a group and how the idea develops, but first, a look at sets and binary operations.

Sets and Binary Operations

One of the fundamental notions in mathematics is that of a set, which is just a collection of things, anything at all, that we are interested in. For example, we could have a set consisting of letters of the English alphabet, A to Z. Or we might just be interested in a part or subset of it, say the vowels or the consonants or just a random collection of some letters or some other kind of objects.

In describing a set, we could list out it’s members or elements explicitly like this:

{A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}

or we could give an expression or rule that generates the elements like this :

{x : x mod 2 = 0}; this is the set-builder notation for the set of numbers that leaves a remainder of 0 when divided by 2; the set of even numbers. For such an infinite set, the set-builder notation saves us an eternity.

A binary operation is a rule for combining two(not necessarily distinct) things to yield a third thing. The things could be numbers or some other kind of entity as long as it makes sense to define the operation on them. The familiar arithmetic operations on numbers(add, subtract, multiply, divide) are examples of binary operations.

So, what is a Group?

Let’s discover it.

Consider a table with a flat rectangular surface. We are going to be working with that rectangular surface not the whole table.

Let’s represent the table surface with the figure below, with the corners marked A,B,C,D.

The dotted lines are lines of symmetry that divide the rectangle into similar parts.

We are interested in ways in which we can move the table corners to new positions, turning the edges, such that it covers the same region of space it covered initially. In other words, we are looking for transformations that maps the object to itself-the symmetry transformations.

It turns out that there are four such transformations:

Transformation 1 (T1) : The trivial, “no-transformation” transformation; just leave the object as it is, or expressed another way, rotate by 0 degrees about the center point-that means no rotation at all. That obviously and easily satisfies the requirement, doesn’t it?

Transformation 2 (T2) : Two right-angle turns about the center point brings back the rectangle to base. A right-angle is 90 degrees, two right-angles is 180 degrees. So Transformation 2 is a 180 degrees rotation.

Transformation 3 (T3) : A flip across the vertical line of symmetry.

Transformation 4 (T4) : A flip across the horizontal line of symmetry.

T1, T2, T3, T4 represent all the symmetry transformations of the rectangle, we can put them in a set like this :

Rectangle symmetry set, S = {T1,T2,T3,T4}; a set with four elements.

We can do the same for other shapes as well; a square being a more symmetrical shape will have more elements in it’s symmetry set, while an isosceles triangle will have less.

CHALLENGE : Find out the symmetry transformations of a square, an isosceles triangle and an equilateral triangle.

So far, we’ve been able to capture and codify the symmetry of the rectangle, next, we’ll like to do what is called composition of transformations and see what pattern emerges.

A composition of transformations is simply one transformation followed by another.

For our rectangle symmetry set above;

T1 followed by T2 still gives T2 as illustrated by the figures below. Doing T1 first then followed by T2 is just like doing T2 only. Get it?

We can regard “followed by” as a binary operation on the symmetry set and represent it by an asterisk *.

It should be obvious that T1 followed by any transformation or vice versa will have no effect on the transformation since it is the “no-transformation” transformation. So, we have:

T1 followed by T1 = T1 or using the operation symbol, T1 * T1 = T1

T1 * T2 = T2

T1 * T3 = T3

T1 * T4 = T4

T1 is the identity element of the set, so named because it doesn’t change the value of the element it operates on, just like multiplication by one.

Now, let’s look at non trivial compositions.

T2 followed by T4 :

T2 is a 180 degree rotation; transforms ABCD to CDAB :

Now, apply T4 to CDAB, remember T4 is a flip across the horizontal line of symmetry :

The final result of T2 followed by T4 is BADC, which is equivalent to applying just T3 on the initial rectangle ABCD, so we can write T2 * T4 = T3. We can represent this result in a Cayley table. A Cayley table, named after Arthur Cayley, a 19th century British mathematician, makes it easy to list out all the possible compositions and their results in a rectangular array.

For the T2 * T4 composition, we have the partially filled Cayley table :

The leading row and the leading column contains the elements to be operated on, the other cells contain the results of the compositions or whatever operation is specified. To compute the result in a cell, take the first operand from the leading column followed by the operand from the leading row.

Filling out the results of the other compositions, we have the complete Cayley table:

Challenge : Convince yourself that the results displayed in the Cayley table above are correct by carrying out the compositions.

From the table we observe that the result of every composition is also a member of the symmetry set we started with, nothing outside {T1,T2,T3,T4}. We say the set is closed under the operation of composition.

A binary operation operates on exactly two elements at a time, what if we need to operate on more than two elements, say something like T2 * T3 * T4; that is an ambiguous expression, to make it meaningful, we can rewrite it as either (T2 * T3) * T4 or T2 * (T3 * T4), and evaluate the expression in brackets first. More complex expressions can be reduced to one those forms.

From the Cayley table above or by direct computation we have :

(T2 * T3) * T4 = T4 * T4 = T1

T2 * (T3 * T4) = T2 * T2 = T1

So the two expressions evaluates to the same result;

(T2 * T3) * T4 = T2 * (T3 * T4) = T1, and if you check you will find that every other such pair of expressions made up of elements from our set are equal to each other. We say the operation * on our set is associative.

Finally, let us look at the inverse of a transformation ; after applying a symmetry transformation to the rectangle, what other transformation cancels the effect and brings it back to the original state(ABCD or T1).

Let’s start with T4 — the flip across the horizontal line of symmetry; transforms ABCD to DCBA. What transforms DCBA back to ABCD? In other words, what transformation satisfies this equation T4 * ? = T1 (the identity element). You can confirm that it’s another flip across the horizontal line of symmetry that cancels out the effect and brings the rectangle back to the identity state. So the inverse of T4 is T4 itself.

Next, what’s the inverse of T3 — the flip across the vertical line of symmetry? Another such flip restores the status quo. So the inverse of T3 is T3.

What about the inverse of T2 — the 180 degrees rotation. Well, the inverse is T2. You can check by turning our table 180 degrees, then another 180 degrees. You’ll come back to the initial state.

Lastly, the inverse of T1 is obviously T1.

So, we have obtained the inverses of all the transformations. These results can also be easily deduced from the Cayley table. The thing to note is that these inverses are members of our set, S.

At this point, we have all the properties that defines a group. We can now give a formal definition of a group.

A group is a set together with a binary operation that has the following properties:

i) Identity. The set has an identity element — the element that has no effect when it operates on another element of the set.

ii) Closure. The result of every operation on the elements yields another element of the set, never outside it.

iii) Associativity. If * is the binary operation, for any three elements T1, T2, T3 of the set we have: T1 * (T2 * T3) = (T1 * T2) * T3

iv) Inverse. Every element has an inverse element in the set.

Always remember that a group consists of two things; a set, and an operation defined on the set. For our rectangle example, the group is made up of the set of symmetry transformations, S and the “followed by” operation we denote by *. In mathematical shorthand, the group is (S,*).

An observation from the above example is that it doesn’t matter if an element comes first or last in an operation, the same result will be produced. T2 * T3 gives the same result as T3 * T2. It applies for all pairs of elements of the set. This is a nice to have property but it is not a requirement in the definition of a group. A group with this additional property is called a commutative group or an Abelian group in honor of Niels Henrik Abel, one of the pioneers of the subject.

Why are we interested in all these? Because the concept of group pops up in a variety of other contexts, it’s like a recurring pattern in nature. Molecules, crystals and many other objects in nature are symmetrical and could be modelled and studied with the language of group theory. Groups also have applications in cryptography and certain puzzles.

Other Examples of Groups

We motivated the concept of a group with transformations in geometry. The idea pops up also in arithmetic; the set of integers(including zero and the negative whole numbers) forms a group with addition(+) as the binary operation. The set of integers is usually represented by Z.

Z = { …-4,-3,-2,-1,0,1,2,3,4,5….}

(Z,+) is a group because :

i) There is an identity element, 0. For any number n in Z; 0 + n = n + 0 = n. Zero added to any element gives back the element.

ii) Z is closed under the ‘+’ operation ; the addition of any two integers gives back an integer.

iii) Addition operation is associative; for any three elements of Z; a,b,c; we have

a + (b + c) = (a + b)+ c, e.g. 2 + (3 + 4) = (2 + 3) + 4 = 9

iv) Any element n of Z has an additive inverse. That inverse is -n. For example the inverse of 2 is -2; 2 + (-2) = 0.

Note that (Z,+) is an abelian group.

Another example from arithmetic is the set of integers modulo 4(the remainders when the integers are divided by 4) denoted by Z₄ . This is a finite group, it has a finite set of elements {0,1,2,3} unlike (Z,+) which is an infinite group. The operation is addition modulo 4, for example 3 + 2 = 1. By regular addition, 3 + 2 = 5, but addition modulo 4 gives 1 because 5 leaves a remainder of 1 when divided by 4. The Cayley table below gives the results of all possible additions in Z₄ and from it we can deduce that (Z₄,+) is a group with 0 as the identity element.

Can we make the alphabet set a group? What sort of operation can we define on letters of the alphabet. Since they are not numbers we can’t perform arithmetic operations on them. How about this as an operation : choose the latter letter and if they are the same just choose one, lets represent it by #

So, G # B = G since G comes latter than B in the alphabet, and C # C = C

Let’s see if the group properties are satisfied.

i) Identity element. If letter A operates on any other letter, we will always get that letter back since A comes before all letters in the alphabet; A # B = B, A # C = C, etc. So, there is an identity element A.

ii) Closure. Obviously, the operation will always produce another letter of the English alphabet.

iii) Associativity. Let’s check with an example. Does B # (G # J ) equals (B # G) # J ? Yes, and the definition of the operation makes that true for any other triplet of letters.

iv) Inverse. Does every letter have an inverse with respect to our defined operation? A is the identity element, let’s find the inverse of B, what letter operates on B to give the identity A? None, since the result of the operation will either be B or a letter that comes after B according to the definition of the operation. This single counterexample is sufficient to show that the inverse property is not satisfied.

So our structure is not a group just because it lacks the inverse property. Not to worry, there is a consolation; such structures are useful and important too, they even have a name; they are called monoids. Now you know how that opening definition comes about.

Group Isomorphism

To illustrate the concept of isomorphism, let’s come up with a made-up group consisting of the set M = {β, φ, Ω, ψ} and the group operation &, and having the following Cayley table :

Compare the rectangle symmetry group, (S,*) and our made-up group (M, &); they have the same number of elements, four each, we can map each and every element in S to an element in M and vice versa; it’s a two-way thing, a bijection.

One of such possible maps is this :

In S, T2 * T3 = T4, according to the map above the avatar of T2 in M is φ, and the avatar of T3 is β, so the parallel of the operation T2 * T3 is φ & β in M. Since T2 * T3 = T4 in S, we naturally expect the result of the parallel operation φ & β to be equal to the avatar of T4, that is, Ω, which happens to be the case(from the Cayley table) and it is true for operation on all the other elements as well, checking from the Cayley table.

A map between two groups such that the result of the operation on any two elements of one group maps to the result of the operation on the avatars of those two elements in the other group is called an isomorphism. If such a map exist between two groups, the groups are said to be isomorphic.

Another possible map from S to M is this:

But this map is not an isomorphism. As an exercise, you can check why.

Isomorphism is a stronger form of the more general concept homomorphism. We use isomorphism when the two groups have the same number of elements so there is a complete pairing between them. Homomorphism is the same thing only that the two groups may not have the same number of elements.

Two groups being isomorphic means they are essentially the same, they may have different kind of sets and operations, but they have the same structure and behavior, from a group theoretic point of view they are the same.

Challenge : Verify that the groups (S, *) and (Z₄, +) are not isomorphic.

Subgroups and Lagrange Theorem

A subgroup is a subset or a portion of a group that is also a group on its own with the same binary operation as the parent group. Every group has it’s identity element as a trivial subgroup just like every number has one as a trivial factor, moreover, every subgroup must include the identity element. Also, the group itself is regarded as a subgroup, just like a number is a factor of itself. So, every group has at least two subgroups automatically — the trivial subgroup and the whole group. From the foregoing, the likely candidates for subgroups of (Z₄, +) are {0}, {0,1} , {0,2,} , {0,3}, {0, 1, 2}, {0, 2, 3}, {0, 1, 3} and {0, 1, 2, 3}. Subsets like {1, 2}, {2,3}, and {1, 2, 3} do not qualify because they don’t contain 0, the identity element of (Z₄, +).

{0} is the set consisting of just the identity element, we claim it is a (trivial) subgroup — you can easily verify that claim. {0, 1, 2, 3} is the whole group, it is of course a group, nothing to prove. We are left with {0,1} , {0,2,} , {0,3}, {0, 1, 2}, {0, 2, 3}, and {0, 1, 3} as the candidates for the proper subgroups of (Z₄, +).

There is a powerful theorem that helps us, among other things, narrow down the list of likely candidates for the subgroups of a group — Lagrange’s theorem, which states :

The order of a subgroup divides the order of the parent group.

The order of a subgroup or group is the number of elements it has. One implication of this theorem is that groups with prime number orders cannot have proper subgroups and for our (Z₄, +) example, we can only have proper subgroups with two elements, since two is the only number that divides four, apart from one and four. So {0, 1, 2}, {0, 2, 3}, {0, 1, 3} are excluded from the list of proper subgroup candidates since three does not divide four, we are left with {0,1} , {0,2,} , {0,3} out of which only one is a subgroup. Challenge : find out which one by testing for group properties in each case. Also find the proper subgroup(s) of (S,*).

There are lots of other interesting things we could say about groups but this is just a first look.