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Posted 21 May 2009

GROUPS  

Groups are mathematical structures that are an abstraction of symmetry.   This article is a brief introduction to groups skewed in the usual abstractmath.org way toward giving qualitative discussions about groups along with a brief look at the actual math involved in groups.  The section on group axioms is a good place to learn about the axiomatic method in a detailed example.

Groups as abstraction of symmetry

Church A is symmetric in shape when viewed from the front.  Church B is not. 

Mathematicians view symmetry as an action rather than a property.  For example, church A has a symmetry that flips the building around the vertical axis up the middle of the bell tower.  This operation called a symmetry because after flipping it the building looks the same in its form (but not in its decoration).  You can flip the right building around the central vertical axis but it won’t look the same in form.  So that operation is not a symmetry of the building.   

That particular kind of symmetry is called a mirror symmetry.

Example: The symmetries of a square

Below are two symmetries r and h of a square.   The letters label the corners of the squares, and they serve to show where the corners go (see alias).   The symmetry r rotates the square by an angle of  clockwise, and h is a mirror symmetry that flips the square around the diagonal shown in red.   Note that these two symmetries are not symmetries of the square-with-labels, but just of the square itself.  It is the square that does not look different when you rotate or flip it.  The labels change. 

You can perform two symmetries one after the other.  The operation of doing one symmetry after another is called composing the symmetries.  If you do r, then h, the result is another symmetry, denoted by hr:

Symmetries can always be represented as functions.  For example, r corresponds to the function  (see barred arrow notation).

Thus hr is the symmetry of flipping about the horizontal center line.  Notice that hr means do r, then h (you perform them right to left).  Some authors write it the other way.  This way of doing it agrees with the usual way of writing composition of functions. 

The tables below show all the symmetries of the square and how they compose.  To find the product of xy from the table, you look at the element in the row labels x and the column labeled y.  Thus the table says that hr = g.  Such tables are called multiplication tables, but remember this multiplication is composition of symmetries and has nothing to do with numbers.

Properties of the symmetries of a square

You can check that all these properties are true of the set of symmetries of the square:

¨ Closure.  Composing two symmetries gives another symmetry.

¨ Existence of identity element.  Doing nothing, the operation e,  is a symmetry.  It has the property that for any symmetry u, eu = e and ue = e.   The operation e is called the identity element of the multiplication.

¨ Associativity.  The multiplication is associative.

¨ Existence of inverses.  Every element has an inverse with respect to e:  For each x there is an element y such that xy = e and yx = e.  The element y is the inverse of x and is denoted by  

¨ Uniqueness of identity.  There is exactly one identity element.

¨ Uniqueness of inverses.  Each element x has exactly one inverse.

All sets of symmetries of a geometric figure have these properties. 

One property that the composition of symmetries of the square does not have is commutativity:   For example,  hr = g but rh = f.   By contrast, an isosceles triangle has just two symmetries and the multiplication is commutative.  So some figures have commuting symmetries and some don’t. 

Axioms for groups

Definition

A group is an algebraic structure  with e a constant (nullary operation), i a unary operation, and  a binary operation, satisfying the following axioms:

1)  For all ,    (e is an identity element)

2)  For all ,    (i(a) is an inverse to a)

3)  For all ,   (associative law)

Other axiom systems for groups

Traditional axiom system

The commonest definition in the math literature uses one binary operation but has two axioms containing existential quantifiers:

A group is a set with a binary operation  for which

1’) There is an element  for which for all , .

2’) For every element  there is an element  for which  

3) For all , .

To show that these axioms are equivalent to 1) through 3) you need to show that 1’), 2’) and 3) imply that there is a unique element of G for which for all ,  and for each  there is a unique element  for which  (then define i(a) = b). 

Weaker axiom system

The identities in the definition above are stronger than they need to be.  You can define a group by requiring only that e be a “left identity” and i(a) be a