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What are groups?

Groups are a fundamental concept in (almost) all fields of modern Mathematics. Here is the modern definition of a group:

A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:

  • Closure: For all a, b in G, the result of a * b is also in G.
  • Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
  • Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.
  • Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.

This is generally the content of the first 5 minutes of the first lecture in the first modern algebra class any student would hear. After this definition is given, the teacher starts describing examples of groups and their properties.

After these 5 minutes, an intelligent student should ask questions like: "Why isn't commutativity assumed" and "Why would all elements have inverses" or even just "Why would we be studying this object in the first place" (in most cases a student would think this but not ask). It is interesting to note that almost any modification to these axioms (by removing some, adding some or replacing some with weaker or stronger versions) gives rise to a named object which which some Mathematician or other has studied at some time. Here is an excerpt from the wikipedia page titled "List of Algebraic Structures" (http://en.wikipedia.org/wiki/List_of_algebraic_structures#Group-like_structures):

  • Magma or groupoid: S is closed under a single binary operation.
    • Implication algebra: a magma satisfying xy.x=x, x.yz=y.xz, and xy.y=yx.x.
    • Steiner magma: A commutative magma satisfying x.xy = y.
      • Squag: an idempotent Steiner magma.
      • Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
    • Semigroup: an associative magma.
      • Equivalence algebra: a commutative semigroup satisfying yyx=x.
      • Monoid: a unital semigroup.
        • Boolean group: a monoid with xx = identity element.
        • Group: a monoid with a unary operation, inverse, giving rise to an inverse element equal to the identity element. thus (b)ba=a holds in all groups.
          • Abelian group: a commutative group. The single axiom yxz(yz)=x suffices.
          • Group with operators: a group with a set of unary operations over S, with each unary operation distributing over the group operation.
          • Algebraic group:
            • Reductive group: an algebraic group such that the unipotent radical of the identity component of S is trivial.
        • Logic algebra: a commutative monoid with a unary operation, complementation, satisfying x(1)=(1) and ((x))=x. 1 and (1) are lattice bounds for S.
          • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
          • Boundary algebra: a logic algebra satisfying (x)x=(1) and (xy)y = (x)y, from which it can be proved that boundary algebra is a distributive lattice. x(1)=(1) and ((x))=x, and xx=x are now provable.
    • Order algebra: an idempotent magma satisfying yx=xy.x, xy=xy.y, x:xy.z=x.yz, and xy.z.y=xz.y. Hence idempotence holds in the following wide sense. For any subformula x of formula z: (i) all but one instance of x may be erased; (ii) x may be duplicated at will anywhere in z.
      • Band: an associative order algebra, and an idempotent semigroup.
        • Rectangular band: a band satisfying the axiom xyz = xz.
        • Normal band: a band satisfying the axiom xyzx = xzyx.

Note that groups are hidden in a forest of magmas, semigroups, monoids and loops. Yet these other objects are not taught (or at most given some brief superficial treatment) in standard algebra classes in universities around the world. Why is this? What is so special about these four specific axioms that make them more "important" than the other similar algebraic structures?


Some possible approaches

Here are some possible explanations that might be given as to why group theory is an important field, more than other group-like algebraic structures.

Elegance of structure

Groups have a beautiful structure theory. For example, a subgroup of a group G is a subset of G that is still closed under the group multiplication (in other words, the subset is a group on its own right). One might ask questions like: Given a finite group G with n elements, and some number k < n, is there a subgroup of G with k elements? The beginning of the answer to this question lies in Lagrange's Theorem (http://en.wikipedia.org/wiki/Lagrange%27s_theorem_%28group_theory%29), a relatively simple result in group theory. The theorem states that the size of any subgroup of a finite group divides the size of the group. This is pretty amazing. Why would an algebraic structure be so strongly linked to numerical properties in this way? This is a great field to study because you encounter so much beauty and structure out of so little assumptions (the axioms are quite permissive). It is results like this do not appear when we demand less (like in semigroups), and if we demand more (like with Abelian groups) we find that the world we are looking at is much smaller (there are "very few" finite Abelian groups -- their structure is completely known, whereas finite groups in general may have very strange and exotic structures).

Applicability

Groups are applicable in many fields in Mathematics and outside of it (Physics and Chemistry, for example). Here is an excerpt from the wikipedia page on Group Theory (http://en.wikipedia.org/wiki/Group_theory#Applications_of_group_theory):

Some important applications of group theory include:

  • Groups are often used to capture the internal symmetry of other structures. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group. Also see automorphism group.
  • Galois theory, which is the historical origin of the group concept, uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The solvable groups are so-named because of their prominent role in this theory. Galois theory was originally used to prove that polynomials of the fifth degree and higher cannot, in general, be solved in closed form by radicals, the way polynomials of lower degree can.
  • Abelian groups, which add the commutative property a * b = b * a, underlie several other structures in abstract algebra, such as rings, fields, and modules.
  • In algebraic topology, groups are used to describe invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups. The name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.
  • The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called harmonic analysis.
  • In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.
  • An understanding of group theory is also important in physics and chemistry and material science. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include: Standard Model, Gauge theory, Lorentz group, Poincaré group
  • In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.
  • Group theory is used extensively in public-key cryptography. In Elliptic-Curve Cryptography, very large groups of prime order are constructed by defining elliptic curves over finite fields.


This makes it an important field to study.

This is a true statement, and might fit well for a course in group theory for physicists, or for people that will be using group theory in the context of its applications in Algebraic Topology for example.

Early History

We have seen that groups are both beautiful (probably more than all other group-like algebraic structures) and applicable (same comment). Is this not a good reason to study them? No. Imagine that we are the first mathematicians discovering and studying groups (a clean mind is a good place to put yourself in whenever you study anything). We don't know about these applications yet, so why would we continue studying these groups? Why did the first group theorists even try to study groups, if they didn't know about the many applications it will have. Why didn't they study loops or monoids or semigroups instead? The same thing is true for the beauty in groups. Before studying them we would not know that we would reach such elegant results. So why groups?

So a natural thing to do (and this is true for any field in Mathematics and outside of it) is to go to history to find out why the first group theorists did what they did. Here is an excerpt from the Wikipedia article on the history of group theory (http://en.wikipedia.org/wiki/Group_theory#History):

There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.

An early source occurs in the problem of forming an mth-degree equation having as its roots m of the roots of a given nth-degree equation (m < n). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.

A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.

Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.

Galois found that if r_1, r_2, \ldots, r_n are the n roots of an equation, there is always a group of permutations of the r's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).

Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.

Walther von Dyck was the first (in 1882) to define a group in the full abstract sense of this entry.

By reading this, you might think that groups are some technical apparatus that helps study solutions to algebraic equations. But this does not give any clue as to how they turn out to be related to so many other fields, or to their apparently special beauty.

As opposed to many cases, where I believe history gives a good answer to the motivation behind a field in Mathematics, for group theory, history shows us only a non-representative small portion of this answer.

The new approach (or: revival of the origins)

For any Mathematical object that has some structure, one might define the concept of an isomorphism. An isomorphism is a function between two objects that preserves the structure of the object completely (meaning the two objects are identical other than giving different names to different elements). Let us illustrate this with some examples:

  1. The sets {1, 2, 3} and {apple, orange, banana} are distinct but isomorphic (rename 1 to apple, 2 to orange and 3 to banana), whereas the sets {1, 2, 3} and {1, 2, 3, 4} are not.
  2. The following two graphs are isomorphic:
    In order to show this, we just need to give names to the vertices of the two graphs so that the relationship between the vertices (whether two vertices are connected by an edge or not) is the same in both graphs: 
  3. The polynomials f(x,y,z) = xy + z and g(x,y,z) = x + yz are isomorphic, since if we transform x to y, y to z and z to x, the polynomial f gets transformed precisely to the polynomial g.

The last graph drawn is highly symmetric. How would we understand this in a precise mathematical way? Let's first think of some a specific symmetry of this graph. Rotating 90 degres clockwise maintains the shape intact. This is essentially the function that brings 1 to 2, 2 to 3, 3 to 4, and 4 to 1. The fact that this rotation maintains the shape intact is essentially saying that this function is an isomorphism from O to itself. An isomorphism from O to itself is called an automorphism of O. Automorphisms of a mathematical object O are the proper mathematical way to discuss "symmetries" of O. This graph has additional symmetries, like rotating 180 degrees (the function that brings 1 to 3, 2 to 4, 3 to 1 and 4 to 2), rotating 90 degrees counter-clockwise (1 to 4, 2 to 1, 3 to 2, 4 to 3), reflecting around the vertical axis like a mirror (1 to 1, 2 to 4, 3 to 3, 4 to 2), and more.

So, essentially, the study of symmetry in any mathematical object is just the study of its set of automorphisms. We call this set Aut(O). In order to study symmetry, we would like to be able to study Aut(O). But right now it seems like the study of Aut(O) will be completely different depending on which object O is. If we would like to discover some general theory of symmetry, we need to find some abstract notion that explains what the different automorphism sets have in common.

In order to discover this, we need to first realize the most important property of automorphisms: They can be composed. This means that if f and g are two automorphisms of O, then we can create a new automorphism which is the end result of having f act on O and then g act on the result. For example, the composition of "rotate 90 degrees clockwise" and "rotate 180 degrees clockwise" is "rotate 90 degrees counter-clockwise", which is also an automorphism. The composition of "rotate 180 degrees" and "reflect around the vertical axis" is "reflect around the horizontal axis" (1 to 3, 2 to 2, 3 to 1, 4 to 4).

So, we can make this abstract: We will say that a set A, together with an operation we call "composition" will be called an "(automorphism set)-like structure". This composition operation should be closed, in the sense that composing two elements of our (automorphism set)-like structure should still stay inside our structure (just like the composition of any two automorphisms is an automorphism). Now we can study these (automorphism set)-like structures and thus discover things about symmetries of all mathematical objects.

In order to do this, we would like to find out what distinguishes arbitrary sets with binary operations defined on them from (automorphism set)-like structures. For example, look at the set {a, b} with the composition operation defined as: a combined with a gives a, a combined with b gives b, b combined with a gives b, b combined with b gives b. Could this be the automorphism set of any object?

...insert some discussion that shows that any such (automorphism set)-like structure must have associativity, inverses and identity. however this is done will not be complete, because the real thing needed is to create a discussion with the student where he explores possibilities and discovers these properties on his own. each student will discover these in a different way...

So, we have "shown" that any (automorphism set)-like structure is a group. But who said there aren't any more properties that we are missing? We now know that the class of groups is large enough to study all symmetries of any mathematical object, but maybe it's too large. For us to believe that it is not too large, we would like a theorem of the sort: For every group G, there is some object O such that G is isomorphic to Aut(O). We have this. Frucht's Theorem, a relatively unknown result, states that any (finite?) group is isomorphic to the automorphism group of some graph. So indeed, the class of groups is precisely the class of objects representing symmetries of any mathematical object (although this indeed we could not know when developing group theory.) This can also be done by polynomials. Any group is the group of symmetries of some polynomial in several variables.

Now that we're done, I found that this is quite nicely explained on Wikipedia: http://en.wikipedia.org/wiki/Group_theory#Connection_of_groups_and_symmetry.

From origin to outcome

It is commonly claimed within the Mathematical community that a theory does not need to have applications in order to be important. To defend this idea, many examples are given of theories that were not applicable at some point and them suddenly became applicable much later. The most extreme case of this is prime numbers, studied first by the Pythagoreans and later became one of the most central concepts of Mathematics -- lately also in "the real world" with cryptography. Another, opposite example would be the theory of perfect numbers, also originated by the Pythagoreans, which has basically died out of modern Mathematics.

Why did prime numbers end up this way and perfect numbers not? And the same goes for groups. Why did groups become this important concept which appears in all fields of Mathematics and monoids or loops not? I claim that the reason is something intrinsic in the concept of a group. Something you could realize even before seeing the applications of group theory, and even before starting to create and theory of groups (during which you'd see the elegant structure theorems). The reason that groups have become what they have become is that they study the symmetry of any mathematical object. So their definition (as given in the new approach in this article) already gives them the strength they need. I believe that Mathematical concepts and theories have intrinsic strength and importance that one can learn to see.

It has been brought to my attention by Vidit Nanda that similarly to the discussion here about automorphisms of objects, one might discuss endomorphisms (homomorphisms of an object to itself) of objects and by this motivating the concept of a monoid. This is true, but still much weaker than the study of automorphisms because the concept of automorphism must exist for any Mathematical object, yet the concept of homomorphism doesn't necessarily exist. For example, it is not at all obvious what a homomorphism of normed vector spaces would be. One definition might be that the norm is preserved, another that the norm gets smaller, another that the mapping preserves the order between norms of vectors, and yet another that the norm of the result is bounded by a constant times the norm of the original vector (which is the "common" definition). Yet many Mathematical objects do have a natural notion of a homomorphism. And indeed, there is a theorem parallel to Frucht's theorem that states that any monoid is isomorphic to the monoid of endomorphisms of a certain graph. This indeed would explain why monoids (and the almost identical semigroups) appear in many fields of Mathematics. Less than groups. More than loops.

--Avital Oliver, avital at thewe dot net

The latest version is always on http://sites.google.com/a/thewe.net/mathematics/What-are-groups-

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