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(2006-02-21)   Monoid
In a monoid, the  associative  internal operator has a neutral element.

Bourbaki  calls "magma" a set endowed with  some  internal operation.  If this operation is  associative  the magma is called a  semigroup.  A semigroup in which there's a neutral element  e  ("x,  ex = xe = x)  is called a  monoid.

Multiplicative notations  are often used where the binary operator is understood between consecutive symbols representing elements.  Applied to monoids or groups,  the qualifier  multiplicative  stresses the use of this convention.

The associative property means the use of parentheses is optional:

x y z   =   (x y) z   =   x (y z)

One rarely bothers endowing a set with a single operation unless it's associative.  However, when two operators are defined, one of them need not be associative.  For example, in the realm of  hypercomplex numbers, the multiplication of octonions or sedenions is not associative.

On the other hand, a monoid operator may or may not be commutative  (there may or may not be pairs of elements for which  xy  and  yx  are different).

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(2006-03-04)   Invertible Elements in a Monoid
Two flavors of invertibility, which coincide when both exist.

In a monoid, an element  x  is said to be  right invertible  if there's a  right-inverse  x'  of  x  (which is to say that the product  xx'  is unity).  It's called   left invertible  if there's a  left-inverse  x''  (such that  x''x  is unity).

When both inverses exist, they are necessarily equal  (HINT:  Consider  x''xx' ). In this case,  x  is said to be  invertible  and its (unique) inverse is denoted  x^-1.

x³  is shorthand for  xxx.  If  x  is invertible,  x^-3  is  x^-1x^-1x^-1.  Note that  x⁰  always denotes the neutral element of multiplication, even when  x  is  not  invertible  (with ordinary arithmetic, zero to the power of zero equals  one).

The Group  M*  of the Invertible Elements of  M :

In a multiplicative monoid  M, the set of all invertible elements form a group which is often denoted  M*.  (It has at least one element, the neutral element of  M.)

For example :

• *  is the multiplicative group of nonzero real numbers. 
• *  is the multiplicative group of nonzero complex numbers. 
• (/n)*  is the finite multiplicative group consisting of the  f(n)  residues modulo n which are coprime with n  (f being Euler's totient function).  This is the structure underlaying Dirichlet's character tables.

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(2006-02-21)   The Free Monoid  (strings over a given alphabet)
A particular monoid where  only  the neutral element is invertible.

All the finite strings (or words) whose characters (letters or symbols) are taken from a given  alphabet  form a monoid under the operation of  concatenation  (concatenating two strings means appending the second to [the right of] the first).  The  empty string  is the neutral element for  concatenation.

This monoid is  free  from any restraints  (equations or conventions)  equating two distinct  strings of symbols.  Hence the name.

Clearly, concatenating two nonempty strings yields something other than the empty string.  As advertised, the empty string is thus the only string with an inverse...

The free monoid over an alphabet of only one symbol is isomorphic to the natural integers endowed with addition  (0,1,2,3...).  In every other case, a free monoid is clearly  not  commutative.

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(2006-02-21)   Group
A group is a  monoid  in which  every  element is invertible.

Walther von Dyck (1856-1934) gave the modern definition of  groups  in 1882.

A group is a set  G  on which an internal operation is defined which verifies the following properties  (using multiplicative notations for the operator).

• Closure :  "xÎG,  "yÎG,   x y  Î G   (The product is "well defined".)
• Associativity :   "xÎG,  "yÎG,  "zÎG,   (x y) z  =  x (y z)
• A  unity element  (e)  exists :   $eÎG,  "xÎG,   e x  =  x e  =  x
• Universal Invertibility :   "xÎG,  $x'ÎG,   x x'  =  x' x  =  e

G  is called a  commutative group  (or  Abelian  group)  when we also have:

• Commutativity (optional) :   "xÎG,  "yÎG,   x y  =  y x

An additive group is merely an Abelian group where additive notations are used;  the "plus" sign  (+)  being used to denote the group operator.

Additive notations are almost never used for a  noncommutative  operator.  The only well-known exception is the "addition" of transfinite ordinals  à la  Cantor  [which we dare regard as a misguided effort].

Single-sided  group  properties imply double-sided ones :

The double-sidedness of two of the above  group axioms  need not be postulated; it can be derived from one-sided equivalents of those axioms :

• There's a  right-neutral  element  e  :    "x,   x e  =  x
• Every element is  right-invertible :   "x, $x',   x x'  =  e

Indeed, we may compute   x' x   using just those two single-sided postulates:

x' x   =   x' x e   =   x' x x' (x' )'   =   x' e (x' )'   =   x' (x' )'   =   e

This will prove  x'  to be the inverse of  x,  if we can establish that  e  is neutral  [on both sides].  That very fact is a consequence of the identity just proven:

"xÎG,   e x   =   (x x' ) x   =   x (x' x)   =   x e   =   x

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This double-sided neutrality implies that there's only one unity  e  (HINT:  Assuming another unity  e', consider  e e' ).

Similarly, there's  only one  inverse  x'  of  x  (HINT:  Let  x"  be another and consider x' x x" ).  So we may safely talk about  the  inverse of x.

This unicity of the inverse shows that   (x' )' = x  (HINT:  x' (x' )' = e ).

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(2006-02-21)   Subgroups
A subgroup is a group contained in another group.

A subgroup H of a group G is a subset H of G which forms a group under the group operation defined over G.  H is a subgroup of G  if and only if  it contains the product of any element of  H  by the  inverse  of any other element of  H.

"xÎH,  "yÎH,    x y^-1 Î H

A  proper  subgroup of G is a subgroup of G not equal to it.

Any  intersection  of subgroups is a subgroup.

The  centralizer  in a group  G  of a subset  E  consists of all the elements of  G  which commute with every element of  E.  It is a subgroup of G.  The centralizer in  G  of  G  itself is the center of  G  (it's the intersection of all centralizers in  G).  The center is a normal subgroup of G, but other centralizers may not be.

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(2006-03-09)   Generators of a Group
The smallest subgroup containing E is said to be generated by E.

For any subset E of a group G, the intersection of all subgroups of G which contain E is a subgroup of G.  It's called the subgroup  generated  by E.

E is said to be a set of  generators  of whatever subgroup it generates.  If there's a finite set E which generates it, the group G is said to be  finitely generated.

For example, the additive group  (,+)  of the integers is generated by the set {1}.  It's also generated by {2,3} or any other pair of coprime integers (because of Bezout's theorem).  More generally, the integers are generated by any set of coprime integers  (not necessarily pairwise coprime)  like  {6,10,15}.

A  finite  group  (of order  n )  which is generated by a single element is a cyclic group.  Each element of such a group which generates the whole group is called a  primitive  element  (or a primitive root, with the vocabulary inherited from representing the cyclic group of order  n  as the "n-th roots of unity" in complex numbers).  There are  f(n)  different elements in a cyclic group which are primitive ones  ( f  being Euler's totient function).

The multiplicative group  (⁺, ´)  of positive rationals is  not  finitely generated.  It's generated by the prime numbers  {2,3,5,7,11,13,17,19...}.

Additive groups which are  not  finitely generated include the rationals, the reals, the complex numbers, the p-adic integers, the p-adic numbers, etc.

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(2006-03-02)   Cosets, Index and Lagrange's Theorem
The order of a subgroup divides the order of the group.

Cosets :

In a group  G,  the  left-coset  of an element  x,  with respect to the subgroup  H,  is the subset  x H  of  G  (consisting of all products   x h   where  h  is an element of  H).  Similarly, the  right-coset  is  H x.

Index of a Subgroup :

Two left-cosets with respect to H are either disjoint or identical and they have the same  cardinality  as H  (i.e., the same number of elements  if  finite).  Whenever it's finite, the number of left-cosets with respect to H is equal to the number of right-cosets.  It's denoted  [G:H]  and is called the  index  of  H  in  G.

Lagrange's Theorem :

In the case of a  finite  group  G,  the fact that such left-cosets form a  partition  of  G  shows that the order of the subgroup  H  divides evenly the order of  G.  (By definition, the  order  of a finite group is its number of elements.)

This result is known as  Lagrange's Theorem.  It is  arguably  the first nontrivial result of  Group Theory.  It's named after Joseph-Louis Lagrange (1736-1813).

Commensurability :

Two subgroups are said to be commensurable when the index of their intersection is finite in each of them.  The qualifier is inherited from ancient Greek mathematics, where two real numbers are called commensurable when they are proportional to two integers.  The two additive groups generated by two such numbers are indeed commensurable in the above sense (their intersection is the additive group generated by the lowest common multiple of the two numbers).

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(2006-03-02)   Normal Subgroups  and  Quotient Groups
The left and right cosets with respect to a normal subgroup are identical.

The concept of a  normal subgroup  is due to  Evariste Galois  (1832).

A subgroup  H  is called  normal  when  aH = Ha  for any a.  Such a subgroup is also called  invariant  or  distinguished  (French:  sous-groupe distingué ).

A subgroup  H  is normal if, and only if, it is stable under  any  inner isomorphism.

"aÎG, "xÎH,    a x a^-1 Î H

Quotient Group of a  Normal  Subgroup :

To a  normal  subgroup H  corresponds an  equivalence relation  among elements of  G  defined by calling x and y equivalent when  xy^-1  is in  H  (in other words, when x and y have the same  left cosets with respect to H).

The  equivalence classes  so defined form a group denoted  G/H  and called the  quotient  of  H  in  G  (or "G modulo H").

Although the above equivalence relation is defined for any subgroup  H,  the equivalence classes form a group  only  when  H  is normal.

Examples of Normal Subgroups :

G  itself and the trivial subgroup  {e}  are normal subgroups of  G,  The derived subgroup  G'  is also always a normal subgroup of  G.

The  center  of a group consists of the elements which commute with  every  element of the group  (such elements are said to be  central).  A  noncentral element is an element which doesn't commute with at least one other element.  The  center  is a normal subgroup.  So is any subgroup of the  center  (in particular, any subgroup of an  Abelian group  is normal).

If  f  is an homomorphism of  G,  then the kernel of  f  (denoted: ker f )  is a normal subgroup of  G.  More generally, so is the reverse image of any normal subgroup of  f (G).  For any normal subgroup  H  of  G,  the direct image  f (H)  is a normal subgroup of  f (G).

For any subset E of the group G, the  subgroup generated by all the conjugates of the elements ofnbsp; E  is called  conjugate closure  of E.  It's a normal subgroup containing E.  In fact, it's the smallest normal subgroup containing E  (i.e, it's the intersection of all normal subgroups containing E).  It's thus also known as the  normal closure  of E.

Any Subgroup is a Normal Subgroup of its Normalizer :

The  normalizer  of a subgroup  H  consists of all elements  x  of the group  G  for which  x H = H x  (in particular all elements of  H  belong to its normalizer).  The normalizer of  H  is a subgroup of  G.  By definition,  H  is a  normal subgroup  of its normalizer  (H  need not be a  normal subgroup  of the whole group G).

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(2006-04-05)   [Group] Homomorphisms
Functions for which the image of a product is the product of the images.

An  homomorphism  is a  map  (or function)  which preserves some specific algebraic operation(s).  A  group homomorphism  is thus a map  f  from a [multiplicative] group  G  into another group  H, which is such that:

"xÎG,  "yÎG,     f(x y)  =  f(x) f(y)

If  f  is surjective  ("onto" H)  it's called an  epimorphism  (or "homomorphism onto").  If it's bijective  ("one-to-one onto")  it's called an  isomorphism.

An homomorphism of G  into itself is called an  endomorphism.  A bijective endomorphism is called an  automorphism.

The  kernel  of an homomorphism is a normal subgroup of  G  defined as:

ker  f   =   { xÎG :  f (x) = e }

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(2006-03-05)   Sym(E) :  The Symmetric Group on  E  
The group of the permutations of  E  (bijections of the set E onto itself).

A  permutation  of  E  is a one-to-one correspondence (bijection) of  E  onto itself.  The term is most commonly used when  E  is finite, but it's also acceptable when  E  is infinite  (possibly uncountably so).

The permutations of  E  are a group under  function composition  (o).

f o g (x)   =   f ( g (x) )

In the finite case, the  symmetric group of degree n  is denoted  S_n.  Its order is the number of permutations of  n  elements, namely  n!  (read "n factorial").

Even  permutations form the  alternating group  A_n  (whose order is  n!/2 ).  The alternating group is the derived subgroup of the symmetric group:  A_n = S'_n

An even permutation is obtained by an even number of switches (swaps of two elements).  The parity, or signature, of a finite permutation may be determined by counting the number of inversions in it.

Cayley's Group Theorem  (1878) :

Arthur Cayley (1821-1895) observed that a group  G  is always isomorphic to a subgroup of  Sym(G).  In the multiplicative group  G,  let's associate to an element  a  the bijection T(a) which sends an element  x  to  ax .  T is an homomorphism, from  G  into  Sym(G),  which is called the  regular representation  of G.

T(a) o T(b)   =   T(a b)

So,  any  finite group of order  n  is isomorphic to a subgroup of  S_n .

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(2006-03-02)   Inn(G):  The Group of Inner Automorphisms on  G
An inner automorphism is a conjugation by a given element of  G.

To any element  a  of  G  is associated a special type of automorphism, called an  inner  automorphism and defined as follows  ( f_a  is called  conjugation by a ).

" x,   f_a(x)  =  a x a^-1       [ Note that  f_a o f_b  =  f_ab ]

Under function composition,  inner automorphisms  form a normal subgroup, denoted  Inn(G), of the group of the automorphisms on G, denoted  Aut(G)  (itself a subgroup of Sym(G), the symmetric group on G).  Conjugation by  a  is the identity function just if  a belongs to the center of  G.  Consequently:

Inn(G)  is isomorphic to the quotient of  G  by its center.

Aut(G) / Inn(G)  =  Out(G)   is the  outer automorphism group  of  G.  Unfortunately, the elements of Out(G) are known as outer automorphisms although they are not "automorphisms" at all !

Note that a subgroup  H  of  G  which is mapped onto itself by  any  inner automorphism is a  normal subgroup  (also called  invariant subgroup).

For example, the above claim that  Inn(G)  is a  normal subgroup  of  Aut(G)  is established by showing that conjugation by  any  automorphism  g  of an inner automorphism  (conjugation by  a)  yields another inner automorphism:

" x,     g o f_a o g^-1 (x)   =   g ( a g^-1(x) a^-1 )   =   g(a) x g(a)^-1   =   f_g(a) (x)

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(2006-03-20)   Conjugates and the  Conjugacy Class Formula
The conjugacy classes of a group G form a partition of G.

Two elements  x  and  y  of a group  G  are said to be  conjugates  when there's an inner automorphism from one to the other, that is, when there's an element  a  of  G  such that  ax = ya.

So defined,  conjugacy  is an equivalence relation (it's reflexive, symmetric and transitive). The  conjugacy class  of an element  x  is the set of all elements of  G  which are conjugate to it.  Every element is in one and only one of those classes  (equivalence classes always form such a  partition). 

If  x  is in the center of G,  denoted Z(G), then the conjugacy class of  x  is  simply  {x}  (a set of only one element).  More generally, we would establish that the number of elements that are conjugate to  x  is equal to the index in  G  of the centralizer  C  of  {x}.  That number is usually denoted  [ G : C ].

Tallying the conjugacy classes with more than one element by assigning a different index  i  to each, we thus obtain the so-called  conjugacy class equation :

| G |   =   | Z(G) |  +  å _i  [ G : C_i ]

The second term can be an empty sum (equal to zero) when G is commutative !

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(2006-03-05)   Simple Groups
A group is  simple  if it does not have any nontrivial normal subgroups.

{e}  and  G  are trivially always  normal subgroups  of  G.  The group  G  is called  simple  if there are no other  normal subgroups  besides those two.

Just like 1 isn't said to be prime, the trivial group  {e}  isn't called "simple".

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(2006-03-06)   Derived Subgroup  G'  (or "Commutator Subgroup")
G',  G⁽¹⁾  or [G,G]  is the subgroup of G generated by its commutators.

The commutator  [x,y]  of two elements of the multiplicative group  G  is:

[x,y]   =   x y x^-1 y^-1

The commutators do not (usually) form a subgroup by themselves.  The  derived subgroup (or commutator subgroup) is the subgroup they generate  (i.e., the smallest subgroup which includes them all).

The derived subgroup of a group is a normal subgroup,  as the following identity demonstrates  (since the set of commutators is thus shown to be stable under  any  inner automorphism, so is the subgroup they generate).

a [x,y] a^-1   =   [ axa^-1, aya^-1 ]

G'  may also be defined as the  smallest  normal subgroup of  G  for which the quotient group  G/G'  is  Abelian  (i.e., commutative).  The group  G/G'  is known as the  Abelianization  of  G  (it's the largest Abelian quotient in G).

Examples of Derived Subgroups :

The derived subgroup of any  Abelian group  is the  trivial  subgroup  (consisting of the neutral element by itself).

The derived subgroup of the symmetric group  S_n  is the  alternating group  A_n.  The derived subgroup of the alternating group is equal to itself:  A'_n = A_n.
The derived subgroup of the Quaternion group  is  {+1,-1}.

Commutator Subgroup   |   Commutator

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(2006-03-21)   Direct Product  (or Direct Sum)

The direct product of two groups  G  and  H  is the group obtained by endowing the cartesian product  G ´ H  with independent operations on the components:

(g,h) (g',h')   =   ( g h , g'h' )

The term  direct sum  may be used for the same concept with additive notations:

(g,h) + (g',h')   =   ( g+h , g'+h' )

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(2006-03-05)   Some Finite Groups
Groups of small orders and their families...

Additive notations  (using the symbol "+" for the internal operator)  are often used for commutative groups  (Abelian groups).  Abelian groups isomorphic to the additive group  C_n = (/n, +)  of residues modulo n are called  cyclic groups.

Cyclic Group  C₅

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

All groups of  prime  order are  cyclic  (as Lagrange's Theorem implies that the subgroup generated by a nonneutral element is equal to the entire group).  The same is true for groups whose order is a  cyclic number  (i.e., an integer coprime to its Euler totient)  according to a result attributed to William Burnside.

The smallest  noncyclic  groups are thus of order  4 and 6...  The so-called  Klein group  is a commutative group of order 4.  The smallest  noncommutative  group is the following group  S₃ = D₃  (the 6 symmetries of an equilateral triangle).

Klein Group

+	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

Dihedral Group  D₃

	A	B	C	D	E	F
A	A	B	C	D	E	F
B	B	C	A	E	F	D
C	C	A	B	F	D	E
D	D	F	E	A	C	B
E	E	D	F	B	A	C
F	F	E	D	C	B	A

The  Klein Group  (V)  is isomorphic to the  direct sum   C2 ´ C2 Felix Klein  called it  Vierergruppe  in 1884.

 

The  dihedral group  D_n consists of the 2n symmetries of a regular n-gon  (n rotations, n flips).

There are 5 groups of order 8.  Three are  Abelian :  C₈  and the two direct sums  C₂+C₄   and  C₂+C₂+C₂  (the additive group of the field of order 8).  The other two groups of order 8 are  noncommutative,  they include the dihedral group D₄  (automorphisms of a square) and the following so-called  quaternion group.

On October 16, 1843, the fundamental equations below  (which imply the multiplication table at right)  occurred at once to Hamilton as he was crossing   Brougham Bridge    (Broom Bridge)  in Dublin.  He carved them into the stone of the bridge  (the original carving is gone but a plaque celebrates this famous act of "mathematical vandalism").

i 2   =   j 2   =   k 2   =   i j k   =   -1		Quaternion Group   Q8   1 i j k -1 -i -j -k 1 1 i j k -1 -i -j -k i i -1 k -j -i 1 -k j j j -k -1 i -j k 1 -i k k j -i -1 -k -j i 1 -1 -1 -i -j -k 1 i j k -i -i 1 -k j i -1 k -j -j -j k 1 -i j -k -1 i -k -k -j i 1 k j -i -1
Red (i) and Blue (j) generators of  Q8

The real line combined with an  oriented  3-dimensional space of basis  (i,j,k)  thus forms the  quaternions,  a  4-dimensional  normed division algebra  similar to the  2-dimensional  complex numbers, except multiplication is  not  commutative:

(a,A) + (b,B)	=	( a+b , A+B )
(a,A)  (b,B)	=	( ab - A.B  ,  aB + bA + A´B )

This is how the 3-dimensional "dot product" and "cross product" were  invented, well before the generalized idea of a vector became commonplace.

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(2006-05-09)   Enumeration of Groups of Small Order
The number g(n) of different groups of order n  (up to isomorphism).

If the integer  n  is coprime with its Euler totient  f(n), then there's only one group of order  n  (the cyclic group).  This applies to the following values of  n:  1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51... (A003277).  This result is attributed to William Burnside (1852-1927) and those numbers are known as  cyclic numbers.

For other orders, the number of distinct groups is given by the following table:

Number of groups of order  n   (A000001)

n	g(n)	n	g(n)	n	g(n)	n	g(n)	n	g(n)	n	g(n)
4 6 8 9 10 12 14 16 18 20 21 22 24 25	2 2 5 2 2 5 2 14 5 5 2 2 15 2	26 27 28 30 32 34 36 38 39 40 42 44 45 46	2 5 4 4 51 2 14 2 2 14 6 4 2 2	48 49 50 52 54 55 56 57 58 60 62 63 64 66	52 2 5 5 15 2 13 2 2 13 2 4 267 4	68 70 72 74 75 76 78 80 81 82 84 86 88 90	5 4 50 2 3 4 6 52 15 2 15 2 12 10	92 93 94 96 98 99 100 102 104 105 106 108 110 111	4 2 2 231 5 2 16 4 14 2 2 45 6 2	112 114 116 117 118 120 121 122 124 125 126 128 129 130	43 6 5 4 2 47 2 2 4 5 16 2328 2 4

g(n) = 2   if  n  is  either  the square of a prime  or  a squarefree number with  only one  of its prime factors congruent to  1  modulo another  (A054395).  The following table gives, for each  m,  the numbers  n  for which  g(n) = m.

Groups of order  2ⁿ  (A000679)   |   The Small Groups Library

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(2006-03-05)   Classification of Finite Simple Groups   (1982)
The final result of the work of many group theorists over many years...

The finite simple  Abelian groups  are just the cyclic groups of prime order.

The classification of  noncommutative  finite simple groups is  much  tougher...  Arguably, the final classification effort started with the 1963 publication of a 255-page proof of the  Odd Order Theorem  (or  Feit-Thompson  theorem)  which implies that all noncommutative simple finite groups are of even order:

Solvability of Groups of Odd Order
by  John G. Thompson (1932-)  and   Walter Feit (1930-2004).
Pacific Journal of Mathematics  13  (1963)   775-1029.

The classification was declared complete in 1982,  despite pending gaps...  This was the result of a tremendous collective effort spanning several decades.  A key figure in this accomplishment was Daniel Gorenstein (1923-1992).

The Classification Theorem :

Unless it's one of the  27  sporadic groups  presented below  (including the  Tits Group,  often dubiously tallied with twisted Chevalley groups)  a  finite simple group  necessarily belongs to one of the following 18 countable families:

• The cyclic groups  C_p  of prime order  p.
• The alternating groups  A_n  of degree  n > 4   ( A₅  is of order  60 ).
• 16  types of Chevalley groups, listed below, each uniformly described in terms of a  finite field  of order  q  (q  is a power of a prime number).

Simple Chevalley Groups   ( u Ù v   denotes the GCD of  u  and  v)

Symbol	Order
An(q) ;   n > 0 (q>3 if n=1)	qn(n+1)/2 (n+1) Ù (q-1) n (qi+1-1)   Õ   i =1
Bn(q) ;   n > 1 Except  B2(2)	qn2 2 Ù (q-1) n (q2i -1)   Õ   i =1
Cn(q) ;   n > 2
Dn(q) ;   n > 3	qn(n-1)  (qn-1) 4 Ù (qn-1) n-1 (q2i -1)   Õ   i =1
E6(q)	q36 (q12-1) (q9-1) (q8-1) (q6-1) (q5-1) (q2-1) / 3 Ù (q-1)
E7(q)	q63 (q18-1) (q14-1) (q12-1) (q10-1) (q8-1) (q6-1) (q2-1) / 2 Ù (q-1)
E8(q)	q120 (q30-1) (q24-1) (q20-1) (q18-1) (q14-1) (q12-1) (q8-1) (q2-1)
F4(q)	q24 (q12-1) (q8-1) (q6-1) (q2-1)
G2(q) Except  G2(2)	q6 (q6-1) (q2-1)
2An(q) ;   n > 1	qn(n+1)/2 (n+1) Ù (q+1) n (qi+1 - (-1)i+1 )   Õ   i =1
2B2(q) q = 2 2m+1 > 2	q2 (q2+1) (q-1)
2Dn(q) ;   n > 3	qn(n-1)  (qn+1) 4 Ù (qn+1) n-1 (q2i -1)   Õ   i =1
3D4(q)	q12 (q8+q4+1) (q6-1) (q2-1)
2E6(q)	q36 (q12-1) (q9+1) (q8-1) (q6-1) (q5+1) (q2-1) / 3 Ù (q+1)
2F4(q) q = 2 2m+1 > 2	q12 (q6+1) (q4-1) (q3+1) (q-1)
2G2(q) q = 3 2m+1 > 3	q2 (q2+1) (q-1)

Chevalley groups  are named after Claude Chevalley (1909-1984) who was one of the key founders (in 1935) of the Bourbaki group.  In 1955, Chevalley found a uniform way to describe Lie groups over arbitrary fields.  With finite fields, this led to what J.H. Conway and others have called  untwisted  Chevalley groups  (they're listed first in the above table,  with unsuperscripted symbols).

The so-called  twisted  Chevalley groups result from two modifications of Chevalley's approach;  one proposed by Steinberg, the other by Suzuki and Ree.

Among these, the highlighted entry  ²F₄(2 ^2m+1 )  is a  simple group  for positive values of m.  For m=0 however, this group is not simple but has a simple normal subgroup of index 2  (its derived subgroup)  known as the  Tits Group  and best classified among  sporadic groups.

Classification of Finite Simple Groups   |   List of Finite Simple Groups   (Wikipedia)

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(2006-03-06)   The 26 or 27 Sporadic Groups
Noncommutative non-alternating  finite simple groups  not of Lie type.

20 of these are related to the largest and most famous of them all,  the  Fischer-Griess Monster.  Six other sporadic groups (highlighted) unrelated to the  Monster  are known as  oddments  or  pariahs.

The  27^th sporadic group is, arguably, the  aforementionned  Tits Group.

Sporadic Notes :

The  Fischer-Griess Monster Group  is also known as  Fischer's Monster, or simply the  Monster Group.  It was predicted independently by Bernd Fischer and Robert L. Griess in 1973.  Griess dubbed it the  Friendly Giant  and constructed it explicitely in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.

The  Leech Lattice  is the densest packing of 24-dimensional hyperspheres  (each touches 196560 others).  Its automorphisms feature a center of order 2.  Modulo that center, they form a simple group called the  Conway Group  (Co₁).

Simon P. Norton gave a construction of the group proposed by Koichiro Harada  (now called the Harada-Norton group).  Norton also proposed the  monstruous moonshine conjecture  with his aforementioned advisor, John H. Conway.

The  Higman-Sims Group  (HS)  is named after Donald G. Higman and Charles C. Sims, who described it jointly in 1968.  It's a subgroup of index 2 in the group of automorphisms of the  Higman-Sims graph  (the strongly-regular graph with 100 nodes of degree 22, where adjacent nodes have no common neighbors and nonadjacent nodes have 6 common neighbors).

The  Hall-Janko Group  (HJ)  is named after Marshall Hall, Jr. and Zvonimir Janko.  It's a subgroup of index 2 in the group of automorphisms of the  Hall-Wales graph  constructed by Hall and D. Wales in 1968 (also called Hall-Janko graph)  namely, the strongly-regular graph with 100 nodes of degree 36, where adjacent nodes have 14 common neighbors and nonadjacent nodes have 12.

The modern quest for a complete list of sporadic groups was launched by the discovery of the first of the Janko Groups  (J₁) by  Zvonimir Janko,  in 1965.

The first sporadic groups  (M₁₁ , M₁₂ , M₂₂ , M₂₃ , M₂₄ )  are subgroups of  M₂₄   discovered between 1861 and 1873 by Emile Mathieu (1835-1890).  M₁₂  was actually proven to be simple by Georg Frobenius (1849-1917)—so we're told.

The Matheticians Involved   |   Monstrous Moonshine Theory  (Wikipedia)

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(2006-03-01)   Classical Groups   (multiplicative subgroups of matrices)
Groups of transformations depending on parameters in a  field.

GL(n,K)  denotes the group of invertible n by n matrices with entries in a field K.  The classical groups listed below are subgroups of  GL(n,K).  When  K  isn't specifed, the field of real numbers  (R)  is understood, except that the field of complex numbers  (C)  underlies the groups denoted  U(n)  and  SU(n)  (note, however, that the "dimension" listed is always the real dimension, which is twice the complex dimension whenever applicable).

A subgroup of  GL(n,K)  is called a  linear representation  (or simply a  representation)  of any group it happens to be isomorphic to.

A*  denotes the  adjoint  of the square matrix  A  (namely, the "conjugate transpose" of a complex matrix, or simply the transpose of a real matrix).  A matrix is said to be  unimodular  if its determinant equals 1.  The letter "S" (for  special) in the symbol of a group indicates its elements are unimodular.

Alternate Notations :

A notation like  GL(Kⁿ)  may also be used instead of  GL(n,K).  This has the great advantage of being consistent with more general symbols like  GL(V)  which apply to a vector space  V  whose dimension  may  be infinite.

On the other hand, when a finite field  is used, GL(n,GF(q))  may be denoted GL(n,q).  A similar convention holds for all the symbols tabulated above.  For example, the first type of Chevalley groups is  PSL(n,q) = A_n(q).

There's no risk of confusion with notations like  O(3,1)  as used below, which refer to a real vector space metrically endowed with 3 spacelike dimensions and 1 timelike dimension,  since we've yet to conceive several dimensions of time  and rarely consider a field of one element.

Some Special Cases :

• The simplest unitary group is the "unit circle" or  circle group  (denoted T)  which is isomorphic to  U(1),  SO(2)  and    / .
• SZ(n,C)  is the cyclic group of order n  (it does "look" cyclic).
• The  Möbius Group  is isomorphic to  PGL(2,C)  and/or  PSL(2,C).

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(2006-04-12)  The Möbius Group  (homographic transformations)
The automorphisms of the  Riemann Sphere  (the  projective line).

An  homographic transformation  f  (also called a  Möbius transformation  or a  fractional linear transformation)  sends a complex number  z  to:

f (z)   =	a z  +  b
	c z  +  d

It's a  [bijective]  transformation  of the  projective line  (the complex plane plus a single "infinity" point  ¥  beyond its horizon, so to speak).  The image of  ¥  is  a/c  (or  ¥  if  c = 0 )  whereas the image of -d/c  (or  ¥  if  c = 0 )  is equal to  ¥.

The Stereographic Projection

Projective Line	Riemann Sphere
È {¥}	(a,b,c)  Π 3   |   a 2 + b 2 + c 2  =  1
¥	(0,0,1)
z   =     a  +  i b    1 - c	(a,b,c)     c ¹ 1
z  =  u + iv	(   2 u   ,   2 v   ,   | z | 2 - 1   )   | z | 2 + 1 | z | 2 + 1 | z | 2 + 1

Automorphic functions  (originally dubbed "Fuchsian functions" by Poincaré, around 1884)  are  meromorphic functions  (i.e., ratios of two  holomorphic functions; analytic functions of a complex variable)  which are invariant under a countable infinity of Möbius transformations.

Video:   Moebius Transformations Revealed  by  Douglas N. Arnold  &  Jonathan Rogness

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(2006-03-01)  The Lorentz Group  O(3,1)  has 4 connected components
Each is isomorphic to the  Restricted Lorentz Group  SO⁺(3,1).

h   =

é ê ê ë

-1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

ù ú ú û

The  Lorentz Group  O(3,1)  is isomorphic to  SL(2,C)  and consists of all  4 by 4  real matrices  A  such that A* h A = 1, where h is the metric matrix for three dimensions of space and one dimension of time.

The  O(3,1)  group has 4 connected components.  Each of these components is  not  simply connected :

SO⁺(3,1)       T[ SO⁺(3,1) ]       P[ SO⁺(3,1) ]       PT[ SO⁺(3,1) ]

SO⁺(3,1)  is the  (6-dimensional)  Restricted Lorentz Group  consisting of the elements of the  Lorentz Group  O(3,1)  which preserve the direction of time and the orientation of space  (boosts and 3D rotations).  In the above, T and P denote, respectively, a reversal of time and an inversion of space  (the latter could be either a mirror symmetry about a plane or a symmetry about a point). 

The symbol  SO(3,1)  would denote the "Special Lorentz Group", the subgroup of the matrices of  O(3,1)  with determinant one  (which is a disconnected "half" of  O(3,1),  not a connected "quarter" of it).

Poincaré Group :

The  Poincaré Group  ISO⁺(3,1)  is the  10-dimensional  inhomogeneous  group of noninverting isometries for 3 dimensions of space and one dimension of time.  It consists of transformations mapping  x  to  Lx+a , where  L  belongs to the above  Restricted Lorentz Group  SO⁺(3,1)  and  a  is some  4-vector.

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(2006-03-21)   Local Symmetries of the Physical Universe :  A Primer
The laws of nature are invariant under a certain group of transformations.

God does arithmetic.
Carl Friedrich Gauss (1777-1855) 

In spite of their respective successes, General Relativity and the Standard Model are known to be imperfect theories, incompatible with each other.  The ultimate laws of physics  (if they exist)  could only incorporate those two as approximations applicable to specific experimental domains  (like  Newtonian mechanics  approximates  Special Relativity  for low speeds).

Nobody knows (yet) exactly what symmetries the  ultimate laws  of nature should have,  but we may ponder the groups of local symmetries underlaying modern mathematical theories of the  4  known physical interactions:

Electromagnetism	U(1)	1
Weak interactions	SU(2)	3
Strong interactions	SU(3)	8
Gravity	ISO+(3,1)	10

Maxwell's unification of electricity and magnetism into electromagnetism has been ultimately construed as the discovery that electrodynamics is invariant under local  phase transformations,  with the simple structure of  U(1).  The classical quantity associated with that symmetry  (by  Noether's theorem )  is simply  electric charge.  The quantum theory of electrodynamics  (quantum electrodynamics, or QED)  has turned out to be the basic paradigm for all subsequent quantum theories of physical interactions.  Essentially, QED describes how  photons  "mediate" the force between  electrons  (or any other charged particles).

The  electroweak theory  is a satisfying unification of electromagnetism and weak interactions under the symmetries of the direct product  SU(2)´U(1).  It was devised in 1967 by  Steven Weinberg  (1933-)  and  Abdus Salam  (1926-1996)  who were awarded a Nobel Prize for this  (in 1979)  together with  Sheldon Glashow  (1932-)  upon whose work they had built.  The group  SU(2)  is isomorphic to 3-dimensional rotations.  The  broken  electroweak symmetry translates into  4  vector bosons:   g  (the photon)  Z⁰,  W⁺  and  W^-.

Broken:  In the jargon of mathematical physics, a symmetry is said to be broken when symmetrical equations have an asymmetrical solution.

The theory of strong interactions is known as  quantum chromodynamics  (QCD).  It's based on an  unbroken  local symmetry with the structure of  SU(3), dubbed  color symmetry  because of a dubious similarity with the rules of color vision  (where 3 primary colors may combine to create a "colorless" sensation).  QCD describes how  gluons  mediate the strong force between  quarks  (or anything else carrying a  color charge, including gluons themselves).  There are 8 different types of gluons, corresponding to the 8 dimensions of SU(3).  In this context,  SU(3)  is often denoted  SU_c(3).  "C" stands for color.

As described by  Albert Einstein's  General Theory of Relativity,  gravity's local symmetry is that of the Poincaré group, which preserves spacetime intervals, as well as the direction of time and the orientation of space.  The  Poincaré group  is 10-dimensional.  However, a gauge field  (the  graviton)  is associated only with the 4 dimensions of spacetime translations.  Suspiciously, no such particle or field is associated with the 6 dimensions corresponding to Lorentz symmetries  (3 dimensions for spatial rotations and 3 dimensions for Lorentz boosts).

The so-called  Standard Model  of particle physicists describes both  strong  and  electroweak  interactions in a theoretical framework whose symmetries are those of the group  SU(2)´U(1)´SU_c(3), which has 12 dimensions.

The model depends on a number of parameters, adjusted to fit experimental data but otherwise unexplained.  Assuming a different local symmetry would impose different restrictions, for better or for worse.  One classical group possessing more dimensions of symmetry  (24)  than the  Standard Model  is  SU(5).

The correct local symmetry of  "strong-electroweak"  interactions would still not determine the masses of the vector bosons involved  (particles of spin 1)  unless more is known about the way such a symmetry is  broken.

A mind-boggling  supersymmetry  across different spins  (SUSY)  is most probably required of any quantum theory  designed  to include gravity in a fully unified quantum theory "of everything":  Supergravity, Superstrings, etc.