Group
Semi
group
A semigroup is an algebraic structure
consisting of a non-empty set equipped with a binary operation (an operation
that takes two elements from the set and produces another element in the set)
that is associative. In other words, for all elements a, b, and c in the set,
the operation satisfies the associative property:
(a * b)
* c = a * (b * c)
In a semigroup, there is no requirement for an identity element or inverses, as is the case in groups.
Quasi
group
Let G be any non-empty set and * is a
binary operation than the structure is called Groupoid or quasi group if the
binary operation in the set G satisfy the commutative property.
a*b =
b*a
A groupoid with identity element is called Loop.
Monoid
A monoid is an algebraic structure that
consists of a non-empty set, a binary operation A monoid is defined by the
following Properties:
Associativity: The binary operation
is associative, meaning that for all elements a, b, and c in the set M, the
following equation holds:
(a * b)
* c = a * (b * c)
Identity Element: There exists an
identity element, often denoted as "e" or "1," in the set M
such that for any element a in M, the following equations hold:
a * e = a (right identity); e * a = a (left identity)
Group
A group is a set G equipped with an
operation (usually denoted as *) that satisfies the following properties:
- Closure: For
all a, b in G, a * b is also in G.
- Associativity: For all
a, b, c in G, (a * b) * c = a * (b * c).
- Identity
Element: There exists an element e in G such that for all a in G, e * a = a * e
= a.
- Inverse
Element: For each element a in G, there exists an element b in G such that a * b
= b * a = e, where e is the identity element.
An Abelian group is a mathematical
set equipped with a binary operation that is both commutative and satisfies the
group axioms, including the existence of an identity element and inverses for
all elements.
Examples of
Groups:
Integers under addition, denoted as (Z, +).
Non-zero real numbers under multiplication, denoted as (R*, ×).
Symmetric group Sn, which represents all permutations of n
elements.
Special orthogonal group SO(3), representing rotations in 3D space.
Integral
power of an element
Given a group G with a binary operation
(usually denoted as *) and an element a in the group, the integral power of a
with exponent n, denoted as aⁿ, is defined as follows:
- ·
If n is a positive integer: aⁿ = a * a * a * ... *
a (n times).
- ·
If n is 0: a⁰ is defined as the identity element
(often denoted as "e" or "1") of the group.
- ·
If n is a negative integer: aⁿ = (a⁻¹)⁻ⁿ, where a⁻¹
is the inverse of a in the group.
In the context of groups, it's important to note that the group's
properties, such as associativity and the existence of an identity element and
inverses, ensure that these integral powers are well-defined and obey the group
axioms.
For example, in the additive group of integers
(denoted as Z with addition as the binary operation), if you take an element 2
and raise it to the power of -3, you get:
2⁻³ = (2⁻¹)³ = (1/2)³ = 1/8
So, in this case, the integral power of 2 with
exponent -3 is 1/8.
Order of an Element
Given a group G with a binary operation
(often denoted as *) and an element a in the group, the order of the element a,
denoted as o(a), is defined as follows:
o(a) = the smallest positive integer n such that aⁿ = e, where
"e" is the identity element of the group.
Here are a few key points to keep in mind about the order of an element
in a group:
- · The order of the identity element is always 1, as
raising it to any power (including 1) results in itself.
- · If there is no positive integer n such that aⁿ = e,
then the order of element a is considered infinite.
- ·
In finite groups, all elements have finite orders.
In infinite groups, some elements may have finite orders, while others may have
infinite orders.
Modulo
System
A modulo system, or modular arithmetic, is a
mathematical system that calculates values based on remainders after division
by a fixed positive integer called the modulus. It is commonly used to handle
cyclic or periodic phenomena, such as time on a clock or calendar calculations.
Example: Suppose we are working with modular
arithmetic modulo 12, often represented as "mod 12." This means our
modulus is 12, and we will perform all calculations based on remainders when
dividing by 12.
·
Addition and Subtraction: In modulo
12, if we add 7 and 9, we will get 16. However, in modular arithmetic modulo
12, we consider the remainder when dividing 16 by 12, which is 4. So, 7 + 9
(mod 12) = 4.
Similarly, if we subtract 5 from 10, we
will get 5. In modular arithmetic modulo 12, 10 - 5 (mod 12) is also equal to
5.
·
Multiplication: Let's
say we want to find 6 multiplied by 8, which equals 48. In modular arithmetic
modulo 12, we consider the remainder when dividing 48 by 12, which is 0. So, 6
* 8 (mod 12) = 0.
Division
Algorithm
The division algorithm is a fundamental
mathematical concept that describes how integers can be divided. It states that
for any two integers, a dividend 'a' and a non-zero divisor 'b,' there exist
unique integers 'q' (the quotient) and 'r' (the remainder) such that:
a =
bq + r, 0 ≤ r < |b|
In other words, when you divide 'a' by
'b,' you can express 'a' as the product of 'b' and 'q,' plus a remainder 'r,'
and the remainder is always less than the absolute value of 'b.'
Greatest Common Divisor (GCD)
The greatest common divisor of two
integers 'a' and 'b' is the largest positive integer that divides both 'a' and
'b' without leaving a remainder. It is often denoted as GCD (a, b). For
example, the GCD of 12 and 18 is 6 because 6 is the largest number that divides
both 12 and 18 evenly.
Prime
Integer
A prime integer is a positive integer
greater than 1 that has exactly two distinct positive divisors: 1 and itself.
In other words, a prime integer cannot be divided evenly by any other positive
integer except 1 and itself. Examples of prime integers include 2, 3, 5, 7, 11,
and so on.
Divisibility
in the Set of Integers
In the set of integers, one integer 'a'
is said to be divisible by another integer 'b' if 'a' can be expressed as a
multiple of 'b' without leaving a remainder. In mathematical notation, if there
exists an integer 'k' such that 'a = kb,' then 'a' is divisible by 'b.' For
example, 15 is divisible by 3 because 15 = 3 * 5.
Residue
Classes Modulo N
For example, in modulo 5, the residue
class [2] modulo 5 consists of integers like -8, -3, 2, 7, 12, and so on, as
they all have the same remainder of 2 when divided by 5.
In modular arithmetic, residue classes
modulo N represent the set of integers that have the same remainder when
divided by a positive integer N, called the modulus. These classes group
integers into equivalence classes based on their remainders after division by
N. The residue class [a] modulo N consists of all integers that leave the same
remainder as 'a' when divided by N. Mathematically:
[a] modulo
N = {x ∈ Z | x ≡ a (mod
N)}
Transformation and permutation group
A 1-1 mapping f of a non-empty set S onto itself is called a
transformation.
Or
A transformation group is a group G
that acts on a set X such that for each group element g in G, there exists a
transformation (or function) φ_g from X to itself, satisfying the following
properties:
- ·
Identity: The
identity element of the group, denoted as e, corresponds to the identity
transformation on X, i.e., φ_e(x) = x for all x in X.
- · Composition: For any
two group elements g and h in G, the composition of the corresponding
transformations is also in the group: φg ◦ φh = φ(g
* h) , where ◦ represents function composition and * denotes the group
operation.
- · Inverse: For each group element g in G, there exists an inverse element g⁻¹ in G such that φg⁻¹ is the inverse of φg, i.e., φg ◦ φg⁻¹ = φe.
A 1-1 mapping f of a finite non-empty set S onto itself is called a
permutation.
Or
The permutation group Sn
consists of all possible permutations of the set {1, 2, 3, ..., n}, and the
group operation is the composition of permutations. The identity element is the
identity permutation (leaving the elements unchanged), and each element in the
group corresponds to a specific permutation of the set.
For example, in S3 (the permutation group of three elements), you have the following elements:
1. Identity permutation: (1 2 3)
2. Transpositions (permutations swapping two elements): (1 2), (1 3), (2 3)
3. Cyclic permutations (cyclically rearranging three elements): (1 2 3), (1 3 2)
Properties of Cyclic Groups:
A cyclic group of order n is isomorphic
to either Z_n (integers modulo n) or Z (integers under addition).
Cyclic
Permutation
A cyclic permutation
is a permutation of a finite set, often represented as a product of disjoint
cycles, where each cycle consists of a sequence of elements, and the elements
within each cycle are rearranged in a cyclic manner, such that one element is mapped
to the next element in the cycle, and the last element is mapped back to the
first element. Cyclic permutations are fundamental in group theory as they
generate cyclic subgroups and provide insight into the structure of permutation
groups.
Informally,
Suppose you have a sequence of elements arranged in a
circle, such as numbers on a clock face or positions in a circular racetrack. A
cyclic permutation is a rearrangement of these elements where each element
moves to the position of the next element, and the last element wraps around to
the first position. This creates a cycle of movements.
For example, consider the sequence 1, 2, 3, 4, and
5. A cyclic permutation of this sequence might look like:
1 → 2 →
3 → 4 → 5 → 1
In this cyclic permutation, each element moves one position to the
right, and the element 1 wraps around to the first position.
Length
of a Cycle
In the context of permutations and cycle notation, the "length of a cycle" refers to the number of elements involved in a specific cycle within a permutation. When a permutation is expressed in cycle notation, it consists of one or more cycles, and each cycle is a sequence of elements. The length of a cycle is determined by counting the number of elements in that cycle.
Transposition
A "transposition" is a
specific type of permutation that exchanges the positions of two elements while
leaving all other elements unchanged. In cycle notation, a transposition is
represented as a cycle of length 2.
For example, consider the transposition (1 3),
which represents a permutation that swaps the positions of elements 1 and 3
while leaving all other elements in their original positions. If applied to a
set of elements, it would result in the elements being rearranged in such a way
that 1 and 3 switch places.
Disjoint
Cycle
In cycle notation, a "disjoint
cycle" is a cycle that does not share any elements with another cycle
within the same permutation. In other words, it represents a rearrangement of
elements where none of the elements involved in one cycle are affected by or
included in any other cycle in the same permutation. Disjoint cycles are often
used to represent permutations compactly.
For example, consider the permutation expressed as
a product of disjoint cycles: (1 2 4) (3 5). In this notation, the cycles (1 2
4) and (3 5) are disjoint because they involve different elements, and there is
no overlap between them.
Odd
Permutation
An "odd permutation" is a
permutation that can be expressed as an odd number of transpositions. In other
words, if you can represent a permutation as the product of an odd number of
pairwise element swaps (transpositions), it is considered an odd permutation.
For example, consider the permutation (1 3 2),
which can be represented as a product of transpositions: (1 2) (2 3). Since it
is expressed as the product of an odd number of transpositions (two
transpositions), it is an odd permutation.
Even
Permutation
An "even permutation" is a permutation that can be expressed
as an even number of transpositions. If a permutation can be represented as the
product of an even number of pairwise element swaps (transpositions), it is
classified as an even permutation.
For example, consider the permutation (1 2 3),
which can be represented as a product of transpositions: (1 2) (1 3). Since it
is expressed as the product of an even number of transpositions (two
transpositions), it is an even permutation.
Homomorphism
A group homomorphism is a function
between two groups that preserves the group operation. More formally, let (G,
*) and (H, •) be two groups. A function φ: G → H is a group homomorphism if for
all elements a and b in G, it satisfies:
φ(a * b)
= φ(a) • φ(b)
In other words, applying the group operation in G and then mapping with
φ is the same as mapping with φ first and then applying the group operation in
H.
Isomorphism
A group isomorphism is a bijective
group homomorphism between two groups. If there exists a group homomorphism φ:
G → H that is also bijective (both injective and surjective), then G and H are
said to be isomorphic. Isomorphic groups have the same group structure, and
they are essentially the same group with different labels for their elements.
Automorphism
A group automorphism is an isomorphism
from a group to itself. In other words, it's a bijective group homomorphism
from a group G to itself. An automorphism preserves the group structure while
rearranging the elements of the same group.
Monomorphism
A group monomorphism is an injective
group homomorphism. If a homomorphism φ: G → H is injective (one-to-one), it is
called a monomorphism. Monomorphisms preserve the group structure and are often
used to embed one group into another.
Epimorphism
A group epimorphism is a surjective group homomorphism. If a homomorphism φ: G → H is surjective (onto), it is called an epimorphism. Epimorphisms indicate that the image of φ covers the entire target group H.
Inner
Automorphism
An inner automorphism is a specific
type of automorphism that arises from conjugation within a group. It is defined
on a particular group G and is denoted by "Int(g)," where
"g" is an element of the group G.
The inner automorphism Int(g) is a
function on G that maps every element "x" in the group to its
conjugate by "g." In other words, it transforms an element
"x" to "g * x * g⁻¹," where "g⁻¹" is the inverse
of "g." Mathematically, for all elements "x" in the group
G:
Int(g)(x)
= g * x * g⁻¹
The set of all inner automorphisms
generated by elements of the group G forms a subgroup of the group's
automorphism group, denoted by Inn(G). Inner automorphisms are important in
understanding the group's internal structure and symmetries.
Kernel
of a Homomorphism
The kernel of a homomorphism is a
subgroup of the domain group that maps to the identity element of the codomain
group under the homomorphism. In other words, it consists of all elements in
the domain group that get sent to the identity element of the codomain group by
the homomorphism.
Let φ: G → H be a homomorphism from
group G to group H. The kernel of the homomorphism, denoted as
"ker(φ)," is defined as:
ker(φ) =
{x ∈ G | φ(x) = eH}
Here, "eH" is the identity element of group H.
Subgroup
A subgroup H of a group G is a subset
of G that is itself a group under the same operation as G. It must satisfy the
group axioms.
Proper
Subgroup
A proper subgroup of a group G is a
subgroup H that is a subset of G and is not equal to G itself.
In other words, H is a proper subgroup of G if H is a subgroup of G, but
H is not the same as G.
Improper Subgroup
An improper subgroup of a group G is a subgroup H that is equal to the entire group G.
In mathematical notation, if H is a subgroup of G, and H = G, then H is
an improper subgroup of G.
Cosets
Left cosets and right cosets of a
subgroup H in G are sets of the form gH = {gh | h in H} and Hg = {hg | h in H}
for some g in G, respectively.
Lagrange's
Theorem
If H is a subgroup of a finite group G,
then the order (number of elements) of H divides the order of G.
Cyclic
Group
A group G is cyclic if there exists an
element g in G such that every element of G can be expressed as a power of g. G
is generated by g.
Symmetric Group
The symmetric group Sn is the group of all permutations of n elements. It is non-abelian for n ≥ 3.
