For my beloved Elke
1. Introduction
Nice to see you here. If you open an arbitrary textbook (especially a German one) on group
theory, you would normally find something like this:
1. Definition. Gegeben sei eine Verknüpfung auf einer Menge M. Zwei Elemente a; b ε M mit
ab = ba nennt man vertauschbar. Sind je zwei Elemente in M vertauschbar, so nennt man M
kommutativ oder abelsch. Man nennt M eine Halbgruppe, falls alle a; b; c ε M das Assoziativgesetz erfullen: ( ab)c = a(bc). Eine Halbgruppe mit neutralem Element nennt man Monoid.
Bemerkung. Das neutrale Element bezeichnen wir oft mit 1 (oder 0, falls wir + als Bezeichnung
fur die Verknupfung wahlen).
2. Defnition. Eine Gruppe ist eine Halbgruppe G mit einem linksneutralen Element e, in der zu
jedem g ε G ein h ε G existiert mit hg = e.
Bemerkung. Daraus folgt leicht, da e neutrales Element und jedes Element in G invertierbar ist.
Die Anzahl der Elemente in G bezeichnet man als Ordnung jGj von G
Even if you speak German, you may not understand this, (nor do I ). My intention is therefore to
give you a clear understanding of what a group is and a short introduction on group theory. I'd
also like you to have fun playing with groups and motivate you to learn more about this wonderful topic. One of the best Introductions is: http://dogschool.tripod.com/ I use it frequently for the
nomenclature and for proofs. Other sources are Wikipedia and Proof Wiki. These sources give
you elegant proofs but they are not so easy to understand.
So what is a group? Let us start with an example : ( most English / American text books do
this). Assume you have a wheel with 6 symbols (here we use numbers as symbols)
If You turn
the wheel by
1 turn to the
left then You
get
0
5
1
4
2
1
0
2
5
3
4
3
Now turn the
wheel by 3
turns to the
left then You
get
4
3
5
2
0
1
Finally turn
it by 2 turns
to the left
then You
get the original wheel
0
5
1
4
2
3
If we look on all possible turns of the wheel ( 0 , 1, 2, 3, 4, 5) we find the following abstract rules:
1
We have a set of elements (here the number of turns of the wheel or if you like the different positions of the wheel or what is here most intuitive the natural numbers 0 to 5 or abstract a, b,
c...) and we have an operation • (or here more intuitive +) or if you like a function or connection
(for convenience we use •).
We call a set and the corresponding operation • a group if the following basic rules are obeyed:
! CLOSURE: If a and b are in the group then a • b is also in the group.
! ASSOCIATIVITY: If a, b and c are in the group then (a • b) • c = a • (b • c).
! IDENTITY: There is an element e of the group such that for any element a of the group
a • e = e • a = a.
! INVERSES: For any element a of the group there is an element a-1 such that
a • a-1 = e and a-1 • a = e
That is it! Any mathematical system that obeys those four rules is a group. The study of systems
that obey these four rules is the basis of GROUP THEORY.
The above rules are similar to the rules for addition of integers. Integers are boring (at least for
non mathematicians). There are a lot of structures obeying the rules of Groups. Therefore
Group theory allows us to solve problems in chemistry, quantum mechanics, and Algebra (Galois theory).
We call the number of elements in a group the Order of the group and abbreviate it with |G|
(here we deal only with finite groups).
The theory does not concern itself with what a and b actually are, nor what the operation symbolized by • actually is. By taking this abstract approach, group theory deals with many mathematical systems at once. Group theory requires only that a mathematical system fulfils the
simple rules above. The theory then seeks to find out properties common to all systems that
obey these few rules.
Check the rules for the above system of turns of a wheel or for the natural numbers 0 to 5
where • is the addition modulo 6.
In our example there is one more rule: a • b = b • a;
Groups which obey this additional rule are called commutative or abelian. Now one more example: Let us have a symmetric triangle and the following set of operations:
Turn by zero, one, two to the left. Invert or mirror by the 3 symmetry axes of the triangle. We
get the following
0
2
1
1
a
0
2
2
b
1
0
0
c
1
1
2
d
2
2
0
e
0
1
f
2
Remark: our set of elements are now the above 6 “operations” on a symmetric triangle and our
“group operation” is our first “operation” a and then operation b.
Check the “group rules” and look for example on c • d = f; and d • c = e;
Obviously we have a non commutative group.
1.1 Permutation Groups (Generalisation of the above example)
A permutation of a set X is any bijective function taking X onto X. The set of all such functions
forms a group under function composition, called the symmetric group on G, and written as
Sym(G). A more intuitive definition is the following. Consider a set of n elements and label these
elements (without loss of generalisation ) 1, 2 , 3….. n. e. g. 1,2,3,4. Then reorder (permute)
these elements in some arbitrary way e.g. 4,1,3,2.
There are exactly n! permutations. Written as functions (function table) we have e.g.
X
Identity
F1 or P1
F2 or P2
F3 or P3
F4 or P4
F5 or P5
1
1
1
2
2
3
3
2
2
3
1
3
1
2
3
3
2
3
1
2
1
1
1
2
3
3
2
1
2
2
3
3
1
Now consider e. g. the function F1
X
F1 or P1
and another Function e.g.
X
F3 or P3
and compose these two functions (execute first the second function and then the first). The result
is:
X
F3 or P3
1
3
2
2
3
1
I guess that it is immediately clear that the set of all bijective functions or permutations of a
given set X form a group.
Let us check the axioms (although I do a bit lazy proofing):
3
! CLOSURE: consecutive execution of two permutations is a permutation (or if you prefer
consecutive execution of two bijective functions is again a bijective function)
! ASSOCIATIVITY: the result of the composition of 3 (or more) permutations is independent where you set the brackets. P1*(P2*P3) = (P1*P2)*P3
! IDENTITY: There is a permutation e with P*e = e*P for every Permutation. It is the permutation which leaves the order of the set X unchanged or the function which maps
every element of X on itself.
! INVERSES: P-1 is the function or permutation which maps each image of x ⊂ X to x or
permutes the set X to the original order.
By the way our S3 is exactly the same group of our second example the cycling and mirroring of
a symmetric triangle.
It is an old trick of most mathematical theories to use the characteristics and properties of a
mathematical field to proof and find characteristics of another field which in the same sense is
related to the first field.
Let us give an example:
Consider the following matrices:
1
Id
1
1
M1
1
1
1
1
1
1
M3
M2
1
1
1
1
M4
M5
1
1
1
1
1
Using the rules for matrix multiplication operating on the set X we get e.g. for the matrix
1
M2
1
X
1
1
2
3
=
2
1
3
the permutation or function P2 F2.
4
1.2 Representation of groups:
The most intuitive and common representation of a group is a “multiplication table” or in the
case of a abelian group a “addition table”. Those tables are called Cayley Tables. We represent
our groups as:
First
example:
second example
or if you like a representation according to the definition of group 2 or as a visual function table
you may use:
Ok, why are we sure that in the multiplication (or addition) table in a row or column there are not
2 or more entries identical. Here is a proof:
Assume: a • b = a • c ; (that means the entries in row a are in column b and c the same). Multiply by a-1 from left and we get: a-1 • a • b = a-1 • a • c ;
That means b = c or column b and column c are the same. This law is called the cancellation
law.
Remark: there are lots of elegant proofs, especially if we weaken the rules (half groups, etc). It
is not my intention to explore all the beauty of groups and related topics. My intention is to motivate you to explore this for yourself and give you some basic understanding for your journey.
5
More representations of groups:
Have a look on the design menu. You will find various representations. Of special interest are
graphical representations e.g. the graph of all internal cycles of a group and the so called
Cayley graphs (here for the S3 and the Z6 with two generators)
6
2 Subgroups
Subgrouping is an essential part of group theory because it helps us to discover the inner structure of a group.
We define:
A Subset S of a group G is a subgroup if for all a, b of S the CLOSURE rule
a • b is member of S is fulfilled ( the operation • is the operation of the group G)
Now we have to check the rules:
Rule 1 is true by definition
Rule 2 is true we use for the “operation” the operation • of the original group.
But what is about Rule 3 and 4. Is there e in the subgroup and a a-1 for every a in the subgroup
with a • a-1 = e and a • e = e • a = a;
Consider the Cayley table for the finite subset. By the cancellation law we know that each element can appear at most once in each row and in each column. Since the subgroup is closed,
each row and column must be just the elements of the subset in some arrangement. More specifically, for a given element a the row and column corresponding to a must contain a. (in our
representation we omit the rows and columns and just show the elements of the table). Thus
some member of the subset multiplies a to give the result a. The only element that could do
this is the identity element e. Thus e must be in the subset. Similarly, if e is in the subset
then it must appear once in the row and once in the column corresponding to a. Therefore
there is some element of the subset which multiplies a to give e. That can only be a-1.
Thus e must be in the subset and for every a in the subset
a-1 must also be in the subset. Therefore the subset is a subgroup of the given group.
Generate Your subgroups using the group Ops Menu.
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2.1 Cosets
For a Element g of G we define the right Coset of a Subgroup S as:
Sg = {s•g | s is in S} (we multiply S by the right with the element g) analogue we define the left
Coset as gS = {g•s | s is in S}
Using this definitions we proof the famous Langrange Theorem. But have before a look on the
various representations in the design menu especially the coset graph menu. Examples for S3.
8
Lagrange Theorem
If S is a subgroup of G then the Order of S ( |S| )divides the Order of G.
First we proof a helpful Lemma:
If Sg = {s•g | s is in S} = { e•g , s1•g, …. , s|S|-1•g } is a right Coset of S (same is true for left
Cosets and please don’t bother if you put s or g to the left but you have to be consequent) then
the number of elements in Sg the same as the number of element in S or the number of elements in Sg is the Order of S. This follows immediately from the cancellation law because for
some g in G the products s•g will give different results for all s in S.
Now we proof:
Two right (or left) Cosets of S are either disjoint or identical.
If that is true, the rest of the proof is trivial because successively we exhaust all elements of G
putting them in some Coset of S or in S (S is the Coset Se with e the neutral element). That
means the Order of S divides the Order of G.
Here is the remaining part of the proof of Lagrange's Theorem.
Assume the Cosets Sg1 and Sg2 have an element in common which means for some elements
s1 and s2 we have s1•g1 = s2•g2 or g1 = s1-1 • s2 • g2 ;
Because s1 and s2 are elements of S the product s3 = s1-1 • s2 is also in S and we may write
g1 = s3 • s2 ;
Which means, we may write all elements s•g1 ( s in S) of Sg1 as s • s3 • g2 or (s • s3) • g2;
but by definition (s • s3) • g2 is Sg2 if we build all products (s • s3) for all s in S. (hint use the cancellation law to proof that the set of elements (s • s3) is the Group S).
Here are some useful definitions:
If S is a Subgroup of G then the number of Cosets of S is called the index of S in G or (G:S).
We may write Lagranges theorem as
|G| = |S| * (G:S)
Lagranges Theorem gives us a first insight in the inner construction rules of groups. One interesting question is:
! If m is a divisor of the order of G, is there a subgroup of G with order m?
Those Questions are dealt in the Sylow Theorems. Here is a simple question from the dog
school:
! If the order of the group G is divisible by 2 then there exists always a subgroup S of G
with |S| = 2;
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2.2 More about cosets.
We used cosets to proof Lagranges theorem. Let us have a closer look on the relation between
left and right cosets. For that we need the concept of conjugation.
Definition
Suppose G is a group. Two elements a and b of G are called conjugate if there exists an element g in G with gag−1 = b. Alternatively you may say b is the conjugation of a with g.
Let us go back to the definition of the right coset of S generated by the element g ∈ G:
Sg = {s•g | s is in S} = { e•g , s1•g, …. , s|S|-1•g }
The left coset is defined by
gS = { g• s | s is in S} = {g•e , g•s1, …. , g•s|S|-1}
Instead of multiplying from the right we could multiply from the left with the conjugation of s by g
i.e. g • s • g-1 because g • s • g-1 • g = g • s ;
You may embed Langranges Theorem in a broader concept of equivalence classes and conjugation classes. I suggest Wikipedia.
10
3 Interesting Subgroups
3.1 The Zentralisator of an element
Let h be an element of the group G. We define the Zh = { h •g• h-1 = g | g is in G} or equivalent
Zh is the subset of G with h • g = g • h for all g in Zh. Zh is a subgroup of S. We only have to
proof that the set Zh is closed (the rest is immediate clear).
Let a, b elements of S. Then h •a• h-1 • h •b• h-1 = h •a • b• h-1 is also in Zh. Remark: the bigger
Zh is the more “abelian” is the element h.
3.2 Normal Subgroups (Wikipedia)
A normal subgroup is a subgroup which is invariant under conjugation by members of the group
of which it is a part. In other words, a subgroup S of a group G is normal in G if and only
if gS = Sg for all g in G (see cosets). Normal subgroups (and only normal subgroups) can be
used to construct quotient groups from a given group.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
Other Definitions:
A subgroup, S, of a group, G, is called a normal subgroup if it is invariant under conjugation;
that is, for each element s in S and each g in G, the element gsg−1 is still in S.
For any subgroup, the following conditions are equivalent to normality. Therefore any one of
them may be taken as the definition (I omit the proofs):
For all g in G, gSg−1 ⊆ S.
For all g in G, gSg−1 = N.
The sets of left and right cosets of S in G coincide.
For all g in G, gS = Sg.
Examples: (for definitions of the examples see below)
The subgroup {e} consisting of just the identity element of G and G itself are always normal
subgroups of G. The former is called the trivial subgroup and if these are the only normal subgroups, then G is said to be simple.
The center of a group is a normal subgroup.
The commutator subgroup is a normal subgroup.
3.2.1 Normal subgroups and homomorphisms
If N is normal subgroup, we can define a multiplication * on cosets by
(a1N)*(a2N) := (a1a2)N.
This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f: G → G/N given by f(a) = aN. The image f(N) consists only of the identity
element of G/N, the coset eN = N.
In general, a group homomorphism f: G → H sends subgroups of G to subgroups of H. Also the
preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e}
in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always
normal and the image f(G) of G is always isomorphic to G/ker(f) (the first isomorphism theorem).
11
In fact this correspondence is a bijection between the set of all quotient groups G/N of G and
the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we have shown that the normal subgroups
are precisely the kernels of homomorphisms with domain G.
3.2.2 Visualisation of Langranges Theorem, Cosets and normal subgroups.
Let me try to visualize a little the hard stuff above (Lagranges Theorem, cosets and normal subgroups, homomorhism a.s.o.). Look at the following Cayley Table of a group G with subgroup S
= { 0, 1, 2}:
We construct a right coset Sg by multiplying the members of the subgroup S with the next group
element which is not in S e.g. element 3 (if there is no such element left then we proofed already Langranges Theorem). Obviously S3 has exactly 3 members due to the cancellation law.
We continue with S4, S5, … until no element is left which is not in the cosets we already generated. Now we have to proof that the cosets are disjunct. Assume e.g. element 11 from coset S5
( we generated it by multiplying element 2 from the subgroup S with element 5 from the group)
12
is identical to some element of another coset e.g. S3. I.e. some s of S multiplied with 3 gives
us 2•5 or as equation: s•3 = 2•5;
Multiply now the equation by 2-1 and we get 2-1•s•3 = 5;
Because 2 and s are members of the subgroup S then 2-1•s is also a member of the subgroup S
and we get that element 5 is a member of the coset S3 in contradiction to our assumption.
Let us have a look at the same group and another subgroup:
Looking on the Cayley table we immediately see that the subgroup {0, 3, 8, 11} is normal because the sets of left and right cosets of S in G coincide (Remark: normally a group or subgroup
is generated by some algorithm and using the underlying algorithm you can decide if a subgroup is a normal group. It is pretty cumbersome to decide if an arbitrary table is really a Cayley
table of a group). The above table shows us that the cosets themselves are a group using the
multiplication rule * : (a1N)*(a2N) := (a1a2)N.
E.g. cosets {4,9,7,2}*{1,5,6,10} = {0,3,8,11}. You may verify this directly by multiplying the
members of the two cosets.
13
3.3 The center of a group.
We look on the elements of a group G which are commutative to all elements of the group and
call this subgroup the “Zentrum” or center Z of G (often referred as Z(G) );
Formally: Z( G ) := { z in G | ∀ g in G : gz = zg }
∀ means for all
Z(G) is a subgroup of because if x and y are elements of Z(G) then x•y is a element from Z(G)
because for every g from G we have:
(x•y)•g = x•(y•g) = x•(g•y) = (x•g)•y = (g•x)•y = g•(x•y)
and if x in G then x-1 is in G because
g = g•x•x-1 = x•g•x-1 = x-1•g = g•x-1 ;
and by definition the neutral element e is a member of Z(G)
Properties:
The Center is a abelian subgroup of G, a normal and characteristic subgroup (see below). If G
is abelian then Z(G) = G. The Center consist exactly of all elements with z -1 gz = e; The
smaller the Center of a group is the less abelian is the group. Example G = KleinsGroup X S3
14
3.4 The Commutator subgroup (Wikipedia)
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.[1][2] For elements g and h of a group G, the commutator of g and h is [g,h] := g-1h-1gh. An element of G of
this form is called a commutator. Note: Not all elements of the commutator subgroup are commutators but they are products of commutators. The commutator [g,h] is equal to the identity
element e if and only if gh = hg, that is, if and only if g and h commute.
The commutator subgroup is important because it is the smallest normal subgroup such that the
quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if
and only if N contains the commutator subgroup. So in some sense it provides a measure of
how far away the group is from being abelian; the larger the commutator subgroup is, the "less
abelian" the group is. Examples:
! The commutator subgroup of the alternating group A4 is the Klein four group.
! The commutator subgroup of the symmetric group Sn is the alternating group An.
This means that the symmetric group in this sense is maximum non abelian or minimum
abelian.
! The commutator subgroup of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is
[Q,Q]={1, −1}.
3.5 The Normalizer of a subgroup
Analog to the commutator which tells us how abelian a group is we look now how “not normal” a
subgroup is. Using the definition of a normal subgroup S := { g•s•g-1 ∈ S | g ∈ G; s ∈ S} we
now look for the set N in G with n•s•n-1 ∈ S for all s∈.S. To proof that N is a subgroup we have
to proof that N is closed and that is true because assume n, m ∈ N then n•m • s • (n•m)-1
should be in s and that is true because n (m s m-1) n-1 = n s´ n-1 ∈ S
The more elements the Normalizer contains the more normal is S. It is clear that if N = G then
S is normal.
3.6 Generating subgroups
An easy way to generate a subgroup of a given group is the following:
Assume g1, g2, … gm are elements of the group G.
Now generate the results of all possible combinations and combinations of the combinations of
these elements (eliminate duplicates). The generated set obviously is closed (it is either the
group itself or a subset of the group) and therefore a subgroup S of G.
Example: (Kleins Group of 4) X (S3) generating set {2; 12}.
3.7 Special case: cycle of a single element
If the generating subset is a single element g1 then the generated subgroup is a cyclic group.
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4 Cayley's Theorem
One of the most powerful tools in mathematics is too transfer proofs, properties and other insights from one mathematical structure to another. A famous example is Cayley's theorem.
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that
every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be
understood as an example of the group action of G on the elements of G.
A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written
as Sym(G).
Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems
which are true for permutation groups are true for groups in general. For a better understanding
of the proof, look for every step on the following picture.
x…………………x…………………………..x
T
f0(x)
=
0*x
f1(x)
=
1*x
f2(x)
=
2*x
f3(x)
=
3*x
f4(x)
=
4*x
f5(x)
=
5*x
For any g of G, consider the function fg : G → G, defined by fg(x) = g*x for all x ∈ G. By the cancellation law fg is a permutation of G and so is a member of Sym(G).
The set K = { fg : g ∈ G } is a subgroup of Sym(G) which is isomorphic to G. The fastest way to
establish this is to consider the function T : G → Sym(G) with T(g) = fg for every g in G. T is
a group homomorphism because (using "•" for composition in Sym(G)):
( fg • fh )(x) = (fg ( fh (x)) = fg (h * x) = g * ( h * x) = ( g * h) * x = f (g * h) (x) ;
for all x in G, and hence:
T (g) • T(h) = fg • fh = f (g * h) = T (g * h) ;
16
The homomorphism T is also injective since T(g) = idG (the identity element of Sym(G)) implies
that g*x = x for all x in G, and taking x to be the identity element e of G yields
g = g*e = e. e of G yields g = g*e = e. Alternatively, T is also injective since, if
g*x=g'*x implies g=g' (by post-multiplying with the inverse of x, which exists because G is a
group). Thus G is isomorphic to the image of T, which is the subgroup K.
T is sometimes called the regular representation of G.
Boring proof: I agree. Lets try another way. In dogschool I found the following proof which at
least I understand immediately:
Let's define a group function. That is, a function whose domain is the elements of a
group G and whose range is also the elements of G. Call this function fa where a is an element
of the group G. The action of this function is to multiply any element of G on the right by a. Thus
for any elements x, y or z of G we have
! fa(x) = xa
! fa(y) = ya
! fa(z) = za
For any other element, say b, of G we can also define fb(x) = xb for all x in G. Now what are
these functions? They are nothing other than permutations on the elements of G. To see this we
only need notice that fa(x) = fa(y) means that xa = ya and by the cancellation law this means
that x = y. Also for any element c of G, c = a(a-1c). That is, there is some element
of G (namely a-1c) that fa sends to c. Therefore for every element a of G the function fa is oneto-one and onto--the exact definition of a permutation.
Now we can ask ourselves, how do these functions behave under composition--under the operation followed by?" The answer is exactly like the elements of G under its group operation. To
see this notice that for any element x of G
fa(fb(x)) = fa(xb) = (xb)a = x(ba)
[Remember, fa(fb(x)) means that fb acts on x first "followed by" fa acting on the result of that.
This is equivalent to fb • fa in our followed-by notation.] The repeated operation of the functions
is equivalent to repeated multiplication on the right by the corresponding group element. The
functions/permutations not only form a group under the operation of composition ("followed by")
but that the group that they form is isomorphic to the original group G. This result is known as:
Cayley's Theorem: Every Group is isomorphic to a group of permutations.
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5 Working with Permutations
Now we can reap the fruits of our work:
All what is true for permutations is true for groups and vice versa.
Notation:
In order to understand permutations better (at least for the following) we write (e.g) a permutation as follows
Original set:
{1 2 3 4 5} up to n numbers
Permutated set:
{3 1 2 5 4}
Which means element 1 of the original set is mapped on element 3 of the permuted set and so
on. Shorthand we write {3 1 2 5 4}
We define:
The support of a permutation are the elements of the set which are changed by the permutation
e.g. the support of {1 4 3 2 5} is 4, 2 and the support of {3 2 1 4 5} is 1, 3
Theorem: If the intersection of the supports of two permutations is empty then the two permutations are commutative.
I guess this is immediately clear (although you may find complicated proofs ) because the two
permutations operate on difference sets.
Theorem: every permutation can be decomposed in cycles. The cycles have distinct support.
OK, lets proof it with an example. I guess the example shows immediately the general proof.
{1 2 3 4 5 }
{3 1 2 5 4 }
We start with the first position of the permutation which sends 1 to 3. This is done by the permutation
{1 2 3 4 5}
{3 2 1 4 5} then we look where the 3. Position goes. It is mapped on 2.
This is done by the permutation
{1 2 3 4 5}
{1 3 2 4 5} the we look where the 2. Position goes. It is mapped on 1.
Continuing this procedure we run into a cycle. In shorthand notation we constructed the first
cycle {1 3 2}
Now look if there are any positions left. In our example this is position 4.
We continue the above procedure and end with the cycle {4 5}. So we can write our permutation
in shorthand as {1 3 2} {4 5}
By construction, the cycles have different supports and therefore they are commutative. Our
construction rule shows us also the following theorem
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Theorem: every permutation can be decomposed in transitions:
We define: a transition is a permutation which exchanges exact 2 elements of the original set.
Proof: Instead of continuing our principle without stop to the end of the cycle we stop at every
step. We get the following decomposition (shorthand):
{1 3} {3 2} {5 4}
Notice: Shuffling a cycle to the right by one position gives us a different presentation of the
same cycle. So the decomposition in cycles is unique but the decomposition in transitions is not
unique. But the following holds:
The number of transitions of a cycle of length n is n – 1. If this number is even we say the parity
of the cycle is even if it is odd we say the parity of the cycle is odd.
Analogous we define the parity of a permutation as the sum of the all transitions modulo 2.
We define: the parity of a permutation is the number of number of all transitions of the permutation.
If we multiply a given permutation with a transition, the parity changes. In shorthand notation
you may say add a transition.
Example: take the permutation
{1 2 3 4 5}
{1 3 2 4 5} or shorthand {1 3 2 4 5} = {1 3} {3 2} {5 4}
The parity of this permutation is 1 (odd) because the number of transitions in the decomposition
is 3. At the end of this short introduction in group theory have a look at the following
application which I found in dogschool.
It is about Samuel Loyd's famous challenge with his 14-15 puzzle back in the 1800's. The puzzle consisted of 15 sliding tiles in a square frame with an empty space which a tile could slide
into. The tiles were numbered from 1 to 15. The task was to get the tiles into the normal order:
1
2
3
4
Sam Loyd offered $1000 to any one
1 2 3 4
5
6
7
8
who could find a way to solve the
5 6 7 8
9
10
11
12
the puzzle from the position shown
9 10 11 12
13
14
15
here
13 15 14
Note that the original pattern is a transposition away from the solution. However, it is impossible
to solve from this position since it would take an odd number of transpositions to go from the 1315-14 position to the solved 13-14-15 position. But the empty square can only return to the
lower right-hand corner after an even number of moves. The simplest way to see this is to imagine a checkerboard pattern of light and dark squares painted beneath the tiles. Then each move
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would change the colour of the background square that is showing from light to dark or from
dark to light.
At the end of this short introduction I hope you want to learn more about group theory. Just type
group theory in Your browser and You will get more than 100 Million entries. I hope also that
You have fun using the app. Enjoy:
! The various representations inclusive the graphs
! See immediately the effects of subgroups especially normal subgroups
! Create your own groups of interest
! Play Sudoku based on group theory
! Rearrange graphs according to group theory and challenge Your intuition
! Create psychedelic effects using timers
! And last but not least create music based on group theory
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