Groups are a fundamental concept in (almost) all fields of modern Mathematics. Here is the modern definition of a group: A group ( -
*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 - 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.
- Reductive group: an algebraic group such that the unipotent radical of the identity component of
- Abelian group: a commutative group. The single axiom
- 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.
- MV-algebra: a logic algebra satisfying the axiom ((
- Boolean group: a monoid with
- Equivalence algebra: a commutative semigroup satisfying
- 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*.
- Rectangular band: a band satisfying the axiom
- Band: an associative order algebra, and an idempotent semigroup.
- Implication algebra: a magma satisfying
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 approachesHere 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 structureGroups have a beautiful structure theory. For example, a ## ApplicabilityGroups 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 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 HistoryWe 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 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 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 ( 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 Galois found that if are the 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 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 - 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.
- The following two graphs are isomorphic:
- 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 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 ...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 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 outcomeIt 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 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- |

Motivation >