Related Links (Outside this Site)

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

Bourbaki  calls magma a set endowed with some  internal well-defined operation. (Avoid the term groupoid  for this, which is now best reservedfor aconcept in category theory.)

Multiplicative notations  are often used where the binary operator isunderstood between consecutive symbols representing elements. Saying that an operationis multiplicative  merely stresses the use of that convention usually supplemented by the optional use of a so-called multiplicative, symbol,  (like a dot,  an x-shaped cross,  a delta or any special ad hoc  symbol) whenever a clear separation between elements is deemed typographically appropriate.

Once a particular operator is so singled out, it's called a multiplication and the qualifier multiplicative  can then be used,  especially todistinguish that from co-existing additive  concepts.

Semigroups

If its operator is associative,  a magma iscalled a semigroup. Associativity  is the property which makes the use of parentheses optional:

x y z   =   (x y) z   =   x (y z)

The  order  of a finite  semigroup is its number of elements. We count two semigroups as distinct  when there's no isomorphism  between them:

A semigroup operator may or may not becommutative. (In a commutative  semigroup,  xy  is the same as  yx for any pair of elements  x  and  y.)

The numbers of distinct noncommutative semigroups are obtained from the above two tables, by termwise subtractions.  Those numbers are always even because suchsemigroups come in pairs linked by an anti-isomorphism.

Lone operators are often designed to be associative. In complex structures with several operators, non-associativity may emerge in a natural way:

For example, in the realm of hypercomplex numbers,the multiplication of octonions  or sedenions  is not associative.

A × (B × C)   =   (A × B) × C  +  B × (A × C)

Monoids :

A semigroup  in which there's a neutral element e is called a monoid :

e ,  x ,   e x   =   xe   =   x

The number of noncommutative monoids is obtained by subtracting the correspondingentries of the above tables.  It's always an even number because such monoids come in pairs linked by an anti-isomorphism.

A monoid can also be defined as acategory witha single object  (the arrows of that category being the elements of the monoid).

(2006-03-04)  
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 (:  Consider  x''xx' ).In this case,  x  is said to be invertible and its (unique) inverse is denoted  x^-1.

TheGroup  M* of the Invertible Elements of  M :

In a multiplicative monoid  M, the set of all invertible elementsform agroup which is often denoted  M*. (The neutral element of  M  belongs to  M*.)

For example :

• *  is the multiplicative group ofnonzero real numbers. 
• *  is the multiplicative group ofnonzerocomplex numbers. 
• (/n)* is the finite multiplicative group consisting of the (n) residues modulo nwhich arecoprime with n ( being Euler'stotient function). That's the structure underlyingDirichlet's character tables.

(2006-02-21)  
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  forma monoid under the operation of concatenation (concatenating two strings means appending the second to [the right of] the first). The concatenation of two strings called  A  and  B is best called  "Abefore B".

The empty string is the neutral element for concatenation.

This monoid is free  from any relations equating distinct  strings of basic symbols.  Hence the name (French: monoïde libre ).

Clearly, concatenating two nonempty strings yields something other than the emptystring.  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 integersendowed with addition  (0,1,2,3...).  In every other case, a free monoidis clearly not  commutative.

(2013-01-02)    (forpower-associative "multiplications")
Raising something to the power of an integer.

Whenever some kind of associative multiplication is defined,  something like  x³  issimply shorthand for  xxx.  It's always legitimate to raisean element to the power of a positive  integer that way.

x⁰  is always defined as equal to the neutral element, if there is one  (otherwise, it's undefined). It makes no difference whether  x  is invertible or not  (with ordinary arithmetic,zeroto the power of zero equals one).

If  x  is invertible,  x^-3  simply denotes (x^-1)³. Only invertible elements can be raised to the power of a negative integer.

Empty sums, empty products, empty intersections, etc.

Raising something to the power of zero is a special case of an empty product. The result of not performing at all some well-defined associative operation dependson that operation alone:  It's equal to its neutral element  (whenever it has one).

• An empty sum is  0  (the neutral element for addition).
• An empty product is  1  (the neutral element for multiplication).
• An emptycategorial product is theterminal element  (if there's one).
• An empty union of subsets is the empty set.
• An empty intersection of subsets is the whole set.
• An empty concatenation of strings  is the empty string.
• The lowest common multiple  of no positive integers is  1.
• The greatest common divisor  of no nonnegative integers is  0.
• Etc.

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

Walthervon Dyck (1856-1934) formally defined groups  in 1882:

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

• Closure : xG, yG,  x y  G  (The product is well-defined.)
• Associativity:  xG, yG, zG,  (x y) z  =  x (y z)
• A unity element  (e)  exists :  eG, xG,  e x  =  xe  =  x
• Universal Invertibility :  xG, x'G,  x x'  =  x' x  = e

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

• Commutativity (optional) :  xG, yG,  x y  =  y x

An additive  group is merely a group (usuallyabelian)  where additive notationsare used:  The plus  sign  (+)  denotes the group operator.

Single-sided group  properties imply double-sided ones :

The double-sidedness of two of the above group axioms  need notbe postulated; it can be derived from one-sided equivalents of thoseaxioms :

• 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' xe  =  x' x x' (x' )'  =  x'e (x' )'  =  x' (x' )'  =  e

That would show  x'  to be the inverse of  x, if we knew that e  is neutral on both sides.  That fact is easy to prove,  using the above as a lemma:

xG,  e x   =   (x x' ) x   =   x (x' x)   =  xe   =   x

---

This double-sided neutrality implies that there's only one unity e . (:  Assume another unity e'  andconsider e e' ).

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

Note, finally, that   (x' )' = x  (: x' (x' )' =e ).

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

A subgroup H of a group G is a subset H of G which formsa group under the group operation defined over G. H is a subgroup of G if and only if  it contains theproduct of any element of  H  by the inverse  of any other element of  H. Amultiplicativesubgroup is said to be stable by division.

xH, yH,   x y^-1 H

When additive  nomenclature and notations are used, this translatesinto the following statement, which says that a subgroup of an additive groupis merely a subset that's stable by subtraction :

xH, yH,   x y H

A proper subgroup of G is a subgroup of G not equal to G itself. The  trivial  group  {e}  has no proper subgroup.

Any intersection  of subgroups is a subgroup.

The centralizer  in a group  G of a subset  E  consists of all the elements of  G  whichcommute with every element of  E. It is a subgroup of G.

The centralizer in  G  of  G  itself is thecenter of  G,  denoted  Z(G) (it's the intersection of all centralizers in  G). The center is anormal subgroup of G, butother centralizers may not be. Elements of  Z(G)  are called central.

(2020-01-27)   A
A multiplicatively absorbent subset of A  is an ideal  of A.

By definition:
For a left-ideal I,the product ax  is inI whenever x is:  aA,aI  I
For a right-ideal I,the product  xa  is inI whenever x is:  aA,Ia  I
Unless otherwise specified,an ideal  is both  a right-ideal and a left-ideal. Note that the empty set is an ideal of any semigroup.

Any intersection of [one-sided] ideals is a [one-sided] ideal. The intersection of all the ideals of a semigroup is called its minimal  ideal. If it's nonempty,  the minimal ideal  M  of a commutative  semigroup is a group. This is to say that  M  has a neutral element, even if the whole semigroup doesn't.

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

For any subset  E  of a group  G,  the intersectionof all subgroups  of  G containing  E  is a subgroup of  G, called the subgroup generated  by  E.

E  is said to be a set of generators  of whatever subgroup it generates. A group which is generated by a finite set 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 lemma). More generally,  (,+) is generated by any set of coprime integers (not necessarily pairwise  coprime)  like  {6,10,15}.

A finite  group (oforder  n )  which is generated by a singleelement is a cyclic  group. An element of such a group which generates the whole groupis called a primitive  element (or aprimitive root,  with the vocabulary inherited from representingthe cyclic group of order  n  asthe "n-th roots of unity" in complex numbers). There are  (n)  different elementsin a cyclic group which are primitive ones (   being Euler'stotient function).

Themultiplicative 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 generatedinclude the rationals,  the reals,  the complexnumbers,  the p-adic integers, thep-adic numbers,  etc.

(2016-05-29)  
Describing a group using the relations obeyed by its generators.

A finitely-generated group can be described by naming a set ofgenerators andstating the nontrivial relations they obey  (the relators). Those relators are normally given by expressions which are equalto the neutral element  (minimally so) but explicit equations are also commonly used.

A free group  has no relators. The simplest free group is isomorphic to the additive group of the integers (,+)  and has thefollowing multiplicative presentation,  which names a single generatorand states no relators:

<a | >

Less trivially,  the octic group D₄ could be presented  as follows:

< r,s  | r ⁴,s ²,srsr  >        or        < r,s  | r ⁴ =s ² =srsr = 1  >

Do not confuse such presentations  with (linear) representations.

(2006-03-02)  
The order of a subgroup divides the order of the group.

By definition, the  order  |G|  of a finite group  G is its number of elements. (The order of an element x is the order of the subgroupgenerated by {x}.)

Cosets :

In a group  G, the left-coset  of an element  x,  with respect tothe 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 :

Twoleft-cosets with respect to Hare either disjoint or identical and they havethe 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 ofright-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 evenlythe order of  G.

This result is known as Lagrange's Theorem. It's now presented as one of the most basic results of Group Theory, named in honor of Joseph-LouisLagrange (1736-1813),who made a related remark in 1777. The general result was probably known to Cauchy (1789-1857) but it was only formally proved in 1861, by Camille Jordan (1838-1922; X1855).

Commensurability :

Two subgroups are said to becommensurable when theindex of theirintersection is finite in each of them. The qualifier is inherited from ancient Greek mathematics,where two real numbers are calledcommensurable when they are proportional totwo integers.  The two additive groupsgeneratedby two such numbers are indeedcommensurable in the above sense(their intersection is the additive group generated by the lowest common multiple of the two numbers).

(2020-05-19)  , 1845)
If aprime number  p  divides |G|, then some element of G has order  p.

The commutative case can be used as a lemma to prove Cauchy's theoremand also its generalization by Sylow  (Sylow's first theorem, 1872).

:  Cauchy's group theorem  holds forabelian groups.

Proof :  Let G be an abelian group G whose order is a multiple of the prime p:  |G| = n = m×p. First,  we see that the proposition is true if G iscyclic, generated by element a  (since the element a^m  is then of order p).

Otherwise, we proceed by induction  on  m, starting with the case  m = 1  which makes p the order of G. This is trivial because,  by Lagrange's theorem, the order of an element must divide the order of the group and canthus only be 1 or p.  In other words, any element of G besides identity is a satisfactoryelement of order p  (which establishes also that G is cyclic).

For  m ≥ 2,  consider an element h  of G, besides identity. Let H be the nontrivial subgroup generated  by h. H is anormal subgroup  (as is any subgroup in the abelian case). Both H and G/H are nontrivial group of order strictly less than n (because we've already disposed of the cases where G is cyclic). Since the product of their orders (respectively |G| and [G:H]) equals n = m×p,  at least one of them is divisible by p. In either case,  the induction hypothesis implies that the corresponding groupcontains an element of order p.  Either way, we can use that to obtainan element of order p in G, as follows:

• If H contains an element x of order p,  then x is also in G and we're done.
   
• If G/H contains an element y of order p,  then y is the class xH of someelement x of G.  For any integer k, the class of x^k is y^k. So, the order of x is the same as the order of y, namely p. 

This concludes the proof that Cauchy's theorem holds for abelian groups.

Proof for non-abelian groups :

(2020-05-19)  
On the number of subgroups of given order in a finite group  G, 

For a prime  p, a p-group  is a group where theorderof any element is a power of  p. If it's a subgroup  of G, it's called a p-subgroup  of G.

In a finite group  G,  a Sylow p-subgroup  (abbreviated p-SSG)  is a maximal  p-subgroup of G. The set of all p-SSG is denoted  Syl_p (G). Remarkably, all of those are isomorphic to each other.

Sylow's First Theorem :

Sylow's Second Theorem :

Sylow's Third Theorem :

(2006-03-02)  
The left and rightcosets withrespect to anormal subgroup are identical.

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

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

A subgroup  H  is normal iff  it's stableunder any inner isomorphism.

aG,xH,  a xa^-1 H

Quotient Group of a Normal  Subgroup :

To a normal  subgroup H  corresponds an equivalence relation  amongelements 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 definedform a group denoted  G/H  andcalled the  quotient of  H  in  G (or of  G  by  H)  also dubbed  "G modulo H".

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

Examples of Normal Subgroups :

Any group  G  is a normal subgroup  of itself (the only non-proper  one).

The trivial group  {e}  is a normal subgroup  ofany group  G  whose neutral element is  e. (It's a proper subgroup of any such  G but itself.)

The derived subgroup  G'  is alsoalways anormal subgroup of  G.

The center  Z(G)  of a group  G  consists of all the elementswhich commute with every  element  G. A member of  Z(G)  is called a central element. A  noncentral  element is an element whichdoesn't commute with at least one other element. The center  is anormal subgroup. So is any subgroup of the center (in particular, any subgroup of an abelian group is normal).

If f  is ahomomorphismor anantihomomorphismfrom group  G,  then thekernelof f  (ker f ) is a normal subgroup of  G. More generally,so is the inverse image  (pre-image) of any normal subgroup of f (G). For a 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  subgroupgenerated byall theconjugates of the elements of  E is called conjugate closure  of E. It's anormal subgroup containing E. In fact, it's the smallest normal subgroup containing E  (i.e, it's the intersectionof all normal subgroups containing E).  It's thus alsoknown 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 asubgroup of  G. By definition,  H  is a normal subgroup of its normalizer (H  need not be a normal subgroup  of the whole group G).

(2019-04-13)  
Notations for  [proper] normal subgroups  (or ideals).

Two conventions are floating around to distinguish between a standard (reflexive) ordering relation  and its strict  (antireflexive) counterpart:

The above notations for normal subgroups  were introduced by Helmut Wielandt  around 1960. They are now also used to denote ideals  in ring theory (since an ideal is to a ring what a normal subgroup is to a group).

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

An homomorphism  is a map (orfunction) 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:

xG, yG,    f(x y)  = f (x)f (y)

If f  issurjective ("onto" H)  it's called an epihomomorphism (or "homomorphism onto"). If it'sinjective ("one-to-one")  it's called an monomorphism.If it'sbijective ("one-to-one onto")  it's an isomorphism.

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

The automorphisms of a group  G  form a group, denoted  Aut(G).

Anti-homomorphisms :

An anti-homomorphism,  with respect to amultiplicative operator, is a function f which reverses the order of that multiplication :

xG, yG,    f (x y)  = f (y)f (x)

In any group, inversion  is an example of an anti-homomorphism:

( x y ) ^-1   =   y ^-1  x ^-1

The concepts defined above for homomorphisms have their counterparts for anti-homomorphisms: Anti-epihomomorphism, anti-monomomorphism, anti-isomorphism, anti-endomorphism  and anti-automorphism.

Kernel   (French: noyau )

For a homomorphism  (or an anti-homomorphism) f  fromgroup  G  to a group of identity  e, the kernel  of f  is a normal subgroup of  G  defined by

ker f   =  { xG  | f (x) = e }

(The homomorphicpre-imageof any  normal subgroup is normal.)

(2006-03-05)    (Gersonides, 1321)
The group of thepermutations 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 (possiblyuncountably so).

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

fg (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!  ("nfactorial").

Log g(n)     (n Log n)^½

Even  permutationsform the alternating group A_n (whose order is  n!/2 ). It's thederived subgroup of the symmetric group: A_n = S'_n

Notations for small permutations :

One standard way to record computations in the realm of very small finite groups is to use a string of different characters  (digits or letters) to denote the permutation which transforms the sorted elements in the top row intothe matching elements of the bottom row.  Both row are placed between parentheses. Juxtaposition of two such notation indicates the composition of the functions so denoted,  with the usual convention that the rightmostfunction is to be applied first  (composition isn't commutative):

Cycle decomposition of a permutations :

A cyclic permutation  of  n  elements is denoted by a sequencebetween parentheses.  The image of an element is the element to its right (the last element is mapped back to the first one).  There are  n equivalent ways to denote such a permutation,  since there's a free choice ofwhich element is written first in the list.  Equivalent notations are equated:

(1 4 5 3 2)   =   (4 5 3 2 1)

A cycle  is a permutation of  n  elements which is a cyclic permutation  of  m  of those elements  (m ≤ n) which leaves the others unchanged.  Two or more cycles are said to be disjoint when operate on different elements  (each cycle applies only to elements which are leftunchanged by the others). Any permutation can be decomposed as a composition of disjoint cycles in a unique way (up to the order of those cycles, which is irrelevant since disjoint cycles commute). Our previous example entail cycles which do not commute because they're not  disjoints.  namely:

(3 4) (2 3)   =   (2 4 3)                 (2 3) (3 4)   =   (2 3 4)

A cycle of order 2  (a 2-cycle)  is called a switch  ora transposition. It's useful to know that the signature of a cycle of order  n  is  (-1)ⁿ⁺¹.

Cayley's Group Theorem  (1878) :

Arthur Cayley (1821-1895)observed that a group  G is always isomorphic to a subgroup of  Sym(G).

Proof :  In themultiplicative group  G, we associate to an element a the bijection  T(a)  which sends an element  x  to ax . T is an injective homomorphism  (i.e.,  a monomorphism) from  G  to  Sym(G), which is called the regular representation  of G.

T(a) T(b)   =   T(a b)

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

(2006-03-02)  
Aninner automorphism is aconjugation by a given element of  G.

To any element a  of  G  is associateda special type ofautomorphism,called an inner  automorphism (French: automorphisme intérieur ) defined as follows ( f_a  is called conjugation by a ).

x,  f_a(x)  = a xa^-1     [ Note that f_af_b = f_ab ]

Under function composition, inner automorphisms  formanormal subgroup  (see proof later in this section) denoted  Inn(G), of the group of the automorphisms on G, denoted  Aut(G) (itself a subgroup of Sym(G),thesymmetric group on G). Conjugation by a is the identity function just if a belongsto thecenter of  G.  Consequently:

Inn(G)  is isomorphic to thequotient of G  by itscenter.

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

More generally,  two subgroups of G are said to be conjugates of each other when there is an inner isomorphism  between them.

---

The above claim that Inn(G)  is a normal subgroup  of  Aut(G) is established by showing thatconjugation by any  automorphism  g  of an innerautomorphism  (conjugation by a) yields another inner automorphism.  That can be proved in a single line:

x,    gf_a g^-1 (x)  =  g (a g^-1(x)a^-1)  =  g(a) x g(a)^-1  =  f_g(a)(x)

(2006-03-02)  
The members of  Out(G)  are classes  of automorphisms of G.

The outer automorphism group  of the group  G is defined as the quotient  of its group of automorphisms by its group of inner automorphisms :

Out(G)  =  Aut(G) / Inn(G)

Unfortunately, the elements of Out(G) are known asouter automorphisms although they're not "automorphisms" at all !

(2014-12-17)  
A concept unrelated  to thecompletenessofmetric oruniform spaces.

A group  G  is said to be centerless when itscenter istrivial,  which is to saythat only the identity element commutes with every element.

A complete group  is a centerless group whose only automorphisms are theinner  ones.  (Equivalently, it's a group whose center and outer automorphism group  are trivial.)

If a group  G  is complete, it's isomorphic to  Aut(G) (itsautomorphisms). However, the converse need not be true  (one counterexample isD₄ ).

(2020-09-14)  
Variously named after Cauchy (1845), Frobenius (1887) or Burnside...

(2006-03-20)  
The conjugacy classes of a group G form apartition of G.

Two elements  x  and  y  of a group  G  are said to be conjugates when there's aninner 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 ofall 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 thecenter 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 conjugateto  x  is equal to theindex in  G of thecentralizer  C  of  {x}. That number is usually denoted  [ G : C ].

Tallying the conjugacy classes with more than one element by assigning eacha different index  i,  we obtain the so-called conjugacy class formula :

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

The second term is an empty sum  (equal to zero)  when G is commutative.

(2006-03-05)  
A group is simple when it has just two normal subgroups.

{e}  and  G  are trivially always normalsubgroups  of  G. The group  G  is said to be simple  when itsonly normal subgroups  are those two.

Just like1 isn't said to be prime,the trivial group  {e}  isn't called "simple".

(2006-03-06)  
G',  G⁽¹⁾  or [G,G] is the subgroup of Ggenerated by its commutators.

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

[x,y]   =   x y x^-1 y^-1  =   x y (y x)^-1

The set of all commutators isn't necessarily a subgroup. What's called the derived subgroup (orcommutator subgroup) is the subgroup theygenerate (i.e., the smallest subgroup which includes all commutators).

The derived subgroup of a group is anormal subgroup, as the following identity demonstrates  (since the set of commutators is thus shownto be stable under any inner automorphism,so is the subgroup theygenerate).

a [x,y]a^-1   =  [axa^-1,aya^-1 ]

G'  is also the smallest  normal subgroupof  G  whose quotient group  in  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.

The derived subgroup of thesymmetric group S_n  is the alternating group A_n. The derived subgroup of the alternating group is itself: A'_n = A_n.

The derived subgroup of theQuaternion group  is  {+1,-1}.

Derived Series :

It's the sequence where the  n+1 st  term is the derived subgroup of the n-th one  (starting with the whole group when  n = 0).

(2006-03-21)  
The group made from the independent juxtaposition of several groups.

The direct product  of two groups  G  and  H  is thegroup obtained by using for the cartesian product G  H independent operations on the components:

(g,h) (g',h')   =   ( g h , g'h' )

The term direct sum  is used for the same concept withadditive notations:

(g,h) + (g',h')   =   ( g+h , g'+h' )

Similar rules can be used for cartesian products of any number of monoids.

Extensions to infinitely many components :

The concept extends naturally to direct sums  (or direct products )  of infinitely many monoids. Such direct sums are usually understood to be finitely restricted (by considering just the elements having only a finite number of componentsthat differ from the relevant neutral element).

For example, thefundamental theorem of arithmeticprovides a standard isomorphism between the multiplicative monoid of the positive integersand the finitely restricted direct sum of infinitely many copies of the nonnegativeintegers  (each such copy being associated with a prime number). Using standard notations,this can be expressed as:

(*, )   =  (

()

, + )

Note that the set appearing in the right-hand-side of the above is countable,because of the parenthesized exponent which indicates a finite restriction in theabove sense.  A lack of parentheses around the exponent would denoteanuncountable setwhich is rarely investigated, if ever (that beast includes elements idenfified with products of infinitely manycoprime integers).

(2016-01-10)  
Finiteabelian groups are either cyclic or direct sums.

Thus, if  n  is the  k-th  power of a prime,the number of non-isomorphic  abelian groups is equal tothe number  p(k)  of partitions  of k.

More generally,if the prime factorization  of  n  is  q₁^k1 q₂^k2... q_m^km  then the number of non-isomorphic abelian groups of order  n  is equal to:

Abel ( n )   =  p( k₁)  p( k₂)  ...  p( k_m)

That's a multiplicative function of  n.  which depends only on its prime signature (A000688).

For what n are there one million abelian groups of order n ?

By trying only the first 61, we see that the only partition numbers which divide  1000000  are  p(1) = 1,  p(2) = 2  and  p(4) = 5. Therefore, there are exactly  1000000 distinct abelian groups of order  n if and only if the factorization of  n  consists of:

• 6  primes of multiplicitity  4.
• 6  primes of multiplicitity  2.
• Any number of primes of multiplicitity  1  (possibly none).

The smallest example is a 35-digit  integer:

(2 . 3 . 5 . 7 . 11 . 13)⁴  (17 . 19 . 23 . 29 . 31 . 37)²  =  4.96597898...³⁴

(2019-04-11)  

(2019-04-11)  

(2014-12-21)  
A semi-direct product of  G  andits group of automorphisms  Aut(G).

If f  is a homomorphism  from a group H  to  Aut(G),  the semi-direct product of  G  and  H  with respect to f  is the group denoted G _f H  consisting ofthecartesian product G  H  with the multiplication :

(x,a)  (y,b)   =  ( xf (a) (y) , ab )

When f  is the trivial homomorphism  (i.e., f (a)  is the identity of  G  for any a) this semi-direct product  is just the direct product  of  G  and  H.

Holomorph :

When  H  is equal to  Aut(G)  we may use the identity of  Aut(G) as the homorphism f  appearing in the above definition and define the holomorph  Hol(G)  as the semi-direct product of  G  and  Aut(G)  in which :

(x,a)  (y,b)   =  ( xa(y) , ab )

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

Additive notations (using the symbol "+" for the operator) are often used for commutative groups  (abelian groups). Groups isomorphic to the group C_n = (/n, +) of residuesmodulo n arecalled 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 (asLagrange's Theorem implies that the subgroupgenerated by a nonneutral elementis equal to the entire group). The same is true for groups whose order is a cyclic number  (i.e., an integercoprimeto itsEuler totient). That result is due toWilliamBurnside.

The smallest noncyclic  groups are thus of order  4 and 6. The Klein group  is the noncyclic 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, nflips).

There are 5 groups of order 8.  Three are abelian : C₈  and the twodirect sums C₂+C₄   and C₂+C₂+C₂ (the additive group of thefield of order 8). The other two groups of order 8 are noncommutative, namely the dihedral groupD₄  (thesymmetries of a square)and the quaternion group Q₈ :

(2006-03-05)  

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 Euclidean space of orthonormal basis  (i,j,k) forms the quaternions,  a  4-dimensional normed division algebra similar to  2-dimensional complex numbers, except multiplicationis not  commutative:

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

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

The above quaternionic units can be used to build a Dirac operator  D (yielding the opposite of the Laplacian   when applied twice):

D    =   i x  + j y  + k z

The Laplacian remains the same in two systems of coordinates (a.ka.reference frames)obtained from each other by rigid rotation.

(2023-03-10)  
The law for multiplication in thealgebra generatedby the Dirac matrices.

The multiplicative group generated by the four gamma matrices a,b,c,d is of order 32. It consists of 16 disjoint pairs of elements which are (additive) opposites of each other. With one element of each such pairs, we form a basis for thealgebra of dimension 16 generated by the 4 gamma matrices.

The group clearly contains the identity matrix of dimension four (I) and the product  e = abcd. Introducing  e  allows every element of the group to be uniquely representedby a sign (+ 1 or -1) along with a signed product of at most two factors among the five elementary elementsa,b,c,d,e.  Withzero such factors, we have 2 elements (+I and -I), with one factor we have 10 elements  (a,b,c,d,e and their opposites) and with 2 distinct factors,we obtain  20 = 2×C (5,2)  elements.  The grand total is indeed 32.

Among the 32 elements of the gamma group, we find:

• 1 element of order 1, namely I.
• 15 elements of order 2:  -I, a, -a, ab, -ab, ac, -ac, ad, -ad, ae, -ae.
• 20 elements of order 4, whose square is -I.

This algebra of dimension 16 is known as the spacetime algebra  Cl (1,3) which is just the Clifford algebra  of dimension 4 with Minkowski metric.

Pseudoscalars (Grade 4) commute only with scalars or bivectors (Grade 2). The bivectors and the scalars form the centralizer  of the pseudoscalars.

The above table doesn't depend on Dirac's representation of  a,b,c,d  in terms of  4×4  matrices. It can be entirely constructed from the pairwise anticommutativity of a,b,c,d and the following relations. Therefore, an isomorphic group is entirely specified by a choice of a basis of four mutually anticommutative elements verifying these:

a² = I ,      b² = c² = d² = -I      and the definition   e = abcd

The last relation implies that  e² = -I   and also that  e  anticommutes with the other four. Thus,  a  is special  (it's the only single-letter element which squares to unity)  but b,c,d,e  are placed on an equal footing.

Here's one remarkable identity:

det ( ta + xb + yc + zd + ue)   =   ( t ² x ² y ² z ² u ²) ²

Enumeratimg the automorphisms of Dirac's gamma group :

Let f  be an automorphism. f (a)  must be of order 2 and is therefore, up to a change of sign,  an element of  {a,ab,ac,ad,ae}. For the three distinct images of b,c,d to anticommute with f (a)  and with each other, they must belong to  {b,c,d,e}  up to sign.  Conversely, if those conditions are met, the images of a,b,c,d generate the whole group,  as the above table can be constructed usingonly the rules for combining single letters.  Thus,  we have  5  choicesfor f (a)  and  4×3×2  choices for the other three letters, knowing that we may then pick any choice of four signs among  16  possibilities. Therefore:

Dirac's Gamma group has  5! 2⁴   =   120 × 16   =   1920   automorphisms.

(2014-12-17)  
The octic group  is represented by the eight symmetries of a square.

This is a centerless group  G  isomorphic to  Aut(G)  but not to  Inn(G). A nice example of an incomplete  group isomorphic to its automorphisms.

The dihedral group D₄ can be represented as the group of the 8 symmetries of a square, with vertices numbered clockwise  1,2,3,4. It'sgenerated by :

• r   =   (2341)   =   Quarter-turn clockwise  (order 4).
• s   =     (42)     =   Flip about a diagonal  (order 2).

Dihedral Group D4    A   B   C   D   E   F   G   H   A  A B C D E F G H  B  B C D A H E F G  C  C D A B G H E F  D  D A B C F G H E  E  E F G H A B C D  F  F G H E D A B C  G  G H E F C D A B  H  H E F G B C D A		s r s   =   r -1   =   (1432)   A   =   e B   =   r C   =   r2 D   =   r3 E   =   s F   =   s r    =   r3 s G   =   s r2   =   r2 s H   =   s r3   =   r s
A  is the identity. C  is the half-turn. B and D  are quarter-turns. E and G  are diagonal flips. F and H  are side flips. Swapping an even  number of the above pairsyields one of the inner automorphisms tabulated at right.		The 4 inner automorphisms  are all even permutations :    A   B   C   D   E   F   G   H   fA = fC  A B C D E F G H fB =fD A B C D G H E F fE =fG A D C B E H G F fF =fH A D C B G F E H

If there was an automorphism swapping an odd number of the threepairs (B,D), (E,G) and (F,H) thenwe could combine it with one of the four inner automorphisms toobtain some automorphism f  leaving  (A,B,C,D)  invariant andswapping either (E,G) or (F,H).  Neither is possible, since:

• If f  only swaps E and G, then  f (B) f (F)  =  B F  =  E   f (B F)  =  G
• If f  only swaps F and H, then  f (B) f (E)  =  B E  =  H   f (B E)  =  F

Therefore, any other automorphism must involve sending at leastone element of the three aforementioned pairs to an element of another.

Any automorphism must leave invariant  A  (the identity)  and  C (the only other element with a square root). Likewise, the order-4 elements, B and D, must be invariant or transform into each other. 

Aut ( D₄ ) , the group of automorphisms of D₄ ,  is isomorphic to D₄.

One of the  8  isomorphisms between D₄ and  Aut ( D₄ )

	D4			Aut ( D4 )			Inn ( D4 )
	A	1234	e		ABCDEFGH	a	fA = fC
	B	2341	r		ABCDFGHE	b
	C	3412	r2		ABCDGHEF	c	fB = fD
	D	4123	r3		ABCDHEFG	d
1	E	1432	s		ADCBEHGF	e	fE = fG
2	F	2143	s r  = r3s		ADCBHGFE	f
3	G	3214	s r2 = r2s		ADCBGFEH	g	fF = fH
4	H	4321	s r3 = r s		ADCBFEHG	h

The  9 propersubgroupsof D₄  areabelian. Seven of them are cyclic (one of order 1, five of order 2, one of order 4)  and two areKlein groups.

The octic group  may also be represented  as a group of 2 by 2 matrices:

A	B	C	D	E	F	G	H
1  0  0  1	0  1  -1 0	-1  0   0 -1	0 -1  1  0	0  1  1  0	-1  0  0  1	0 -1  -1  0	1  0  0 -1
e	r	r2	r3	s	s r	s r2	s r3

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

If the integer  n  iscoprime with itsEuler totient (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 toWilliamBurnside (1852-1927) and those numbers are known as cyclic numbers.

For noncyclic orders (A060679) here's the number of distinct groups:

Number ofgroups 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.

Smallest n for which there are exactly m groups of order n (A046057):
1, 4, 75, 28, 8, 42, 375, 510, 308, 90, 140, 88, 56, 16, 24, 100, 675...

The convention would be to have a zero term to indicate that there's no such n but it's conjectured  that this never happens.

(2006-03-05)     (1982)
The final result of the work ofmanygroup theorists overmany years...

The finitesimple abelian groups  are just thecyclic groups of prime order.

The classification of noncommutative finite simple groups is much  tougher... Arguably, the final classification effort started with the 1963 publication ofa 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 JohnG. 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 decades. A key figure in this accomplishment wasDanielGorenstein (1923-1992).

The Classification Theorem :

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

• Thecyclic groups C_p  of prime order  p.
• Thealternating groups A_n  of degree n > 4   (A₅  is of order  60 ).
• 16  types of Chevalley groups, listed below, eachuniformly described in terms of a finite field of order  q  (q  being the power of a prime). For example, the first such type consists of the projective group of square matrices of dimension  n+1  with coefficients in F_q :

A_n(q)   =   PSL (n+1,F_q)