Discrete Mathematics - Group Theory

Semigroup

A finite or infinite set $‘S’$ with a binary operation $‘omicron’$ (Composition) is called semigroup if it holds following two conditions simultaneously −

Closure − For every pair $(a, b) in S, :(a omicron b)$ has to be present in the set $S$.

Associative − For every element $a, b, c in S, (a omicron b) omicron c = a omicron (b omicron c)$ must hold.

Example

The set of positive integers (excluding zero) with addition operation is a semigroup. For example, $ S = lbrace 1, 2, 3, dots brace $

Here closure property holds as for every pair $(a, b) in S, (a + b)$ is present in the set S. For example, $1 + 2 = 3 in S]$

Associative property also holds for every element $a, b, c in S, (a + b) + c = a + (b + c)$. For example, $(1 + 2) + 3 = 1 + (2 + 3) = 5$

Monoid

A monoid is a semigroup with an identity element. The identity element (denoted by $e$ or E) of a set S is an element such that $(a omicron e) = a$, for every element $a in S$. An identity element is also called a unit element. So, a monoid holds three properties simultaneously − Closure, Associative, Identity element.

Example

The set of positive integers (excluding zero) with multippcation operation is a monoid. $S = lbrace 1, 2, 3, dots brace $

Here closure property holds as for every pair $(a, b) in S, (a imes b)$ is present in the set S. [For example, $1 imes 2 = 2 in S$ and so on]

Associative property also holds for every element $a, b, c in S, (a imes b) imes c = a imes (b imes c)$ [For example, $(1 imes 2) imes 3 = 1 imes (2 imes 3) = 6$ and so on]

Identity property also holds for every element $a in S, (a imes e) = a$ [For example, $(2 imes 1) = 2, (3 imes 1) = 3$ and so on]. Here identity element is 1.

Group

A group is a monoid with an inverse element. The inverse element (denoted by I) of a set S is an element such that $(a omicron I) = (I omicron a) = a$, for each element $a in S$. So, a group holds four properties simultaneously - i) Closure, ii) Associative, iii) Identity element, iv) Inverse element. The order of a group G is the number of elements in G and the order of an element in a group is the least positive integer n such that an is the identity element of that group G.

Examples

The set of $N imes N$ non-singular matrices form a group under matrix multippcation operation.

The product of two $N imes N$ non-singular matrices is also an $N imes N$ non-singular matrix which holds closure property.

Matrix multippcation itself is associative. Hence, associative property holds.

The set of $N imes N$ non-singular matrices contains the identity matrix holding the identity element property.

As all the matrices are non-singular they all have inverse elements which are also nonsingular matrices. Hence, inverse property also holds.

Abepan Group

An abepan group G is a group for which the element pair $(a,b) in G$ always holds commutative law. So, a group holds five properties simultaneously - i) Closure, ii) Associative, iii) Identity element, iv) Inverse element, v) Commutative.

Example

The set of positive integers (including zero) with addition operation is an abepan group. $G = lbrace 0, 1, 2, 3, dots brace$

Here closure property holds as for every pair $(a, b) in S, (a + b)$ is present in the set S. [For example, $1 + 2 = 2 in S$ and so on]

Associative property also holds for every element $a, b, c in S, (a + b) + c = a + (b + c)$ [For example, $(1 +2) + 3 = 1 + (2 + 3) = 6$ and so on]

Identity property also holds for every element $a in S, (a imes e) = a$ [For example, $(2 imes 1) = 2, (3 imes 1) = 3$ and so on]. Here, identity element is 1.

Commutative property also holds for every element $a in S, (a imes b) = (b imes a)$ [For example, $(2 imes 3) = (3 imes 2) = 3$ and so on]

Cycpc Group and Subgroup

A cycpc group is a group that can be generated by a single element. Every element of a cycpc group is a power of some specific element which is called a generator. A cycpc group can be generated by a generator ‘g’, such that every other element of the group can be written as a power of the generator ‘g’.

Example

The set of complex numbers $lbrace 1,-1, i, -i brace$ under multippcation operation is a cycpc group.

There are two generators − $i$ and $–i$ as $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$ and also $(–i)^1 = -i, (–i)^2 = -1, (–i)^3 = i, (–i)^4 = 1$ which covers all the elements of the group. Hence, it is a cycpc group.

Note − A cycpc group is always an abepan group but not every abepan group is a cycpc group. The rational numbers under addition is not cycpc but is abepan.

A subgroup H is a subset of a group G (denoted by $H ≤ G$) if it satisfies the four properties simultaneously − Closure, Associative, Identity element, and Inverse.

A subgroup H of a group G that does not include the whole group G is called a proper subgroup (Denoted by $H < G$). A subgroup of a cycpc group is cycpc and a abepan subgroup is also abepan.

Example

Let a group $G = lbrace 1, i, -1, -i brace$

Then some subgroups are $H_1 = lbrace 1 brace, H_2 = lbrace 1,-1 brace$,

This is not a subgroup − $H_3 = lbrace 1, i brace$ because that $(i)^{-1} = -i$ is not in $H_3$

Partially Ordered Set (POSET)

A partially ordered set consists of a set with a binary relation which is reflexive, antisymmetric and transitive. "Partially ordered set" is abbreviated as POSET.

Examples

The set of real numbers under binary operation less than or equal to $(le)$ is a poset.

Let the set $S = lbrace 1, 2, 3 brace$ and the operation is $le$

The relations will be $lbrace(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3) brace$

This relation R is reflexive as $lbrace (1, 1), (2, 2), (3, 3) brace in R$

This relation R is anti-symmetric, as

$lbrace (1, 2), (1, 3), (2, 3) brace in R and lbrace (1, 2), (1, 3), (2, 3) brace ∉ R$

This relation R is also transitive as $lbrace (1,2), (2,3), (1,3) brace in R$.

Hence, it is a poset.

The vertex set of a directed acycpc graph under the operation ‘reachabipty’ is a poset.

Hasse Diagram

The Hasse diagram of a poset is the directed graph whose vertices are the element of that poset and the arcs covers the pairs (x, y) in the poset. If in the poset $x < y$, then the point x appears lower than the point y in the Hasse diagram. If $x<y<z$ in the poset, then the arrow is not shown between x and z as it is imppcit.

Example

The poset of subsets of $lbrace 1, 2, 3 brace = lbrace emptyset, lbrace 1 brace, lbrace 2 brace, lbrace 3 brace, lbrace 1, 2 brace, lbrace 1, 3 brace, lbrace 2, 3 brace, lbrace 1, 2, 3 brace brace$ is shown by the following Hasse diagram −

Linearly Ordered Set

A Linearly ordered set or Total ordered set is a partial order set in which every pair of element is comparable. The elements $a, b in S$ are said to be comparable if either $a le b$ or $b le a$ holds. Trichotomy law defines this total ordered set. A totally ordered set can be defined as a distributive lattice having the property $lbrace a lor b, a land b brace = lbrace a, b brace$ for all values of a and b in set S.

Example

The powerset of $lbrace a, b brace$ ordered by subseteq is a totally ordered set as all the elements of the power set $P = lbrace emptyset, lbrace a brace, lbrace b brace, lbrace a, b brace brace$ are comparable.

Example of non-total order set

A set $S = lbrace 1, 2, 3, 4, 5, 6 brace$ under operation x spanides y is not a total ordered set.

Here, for all $(x, y) in S, x | y$ have to hold but it is not true that 2 | 3, as 2 does not spanide 3 or 3 does not spanide 2. Hence, it is not a total ordered set.

Lattice

A lattice is a poset $(L, le)$ for which every pair $lbrace a, b brace in L$ has a least upper bound (denoted by $a lor b$) and a greatest lower bound (denoted by $a land b$). LUB $(lbrace a,b brace)$ is called the join of a and b. GLB $(lbrace a,b brace )$ is called the meet of a and b.

Example

This above figure is a lattice because for every pair $lbrace a, b brace in L$, a GLB and a LUB exists.

This above figure is a not a lattice because $GLB (a, b)$ and $LUB (e, f)$ does not exist.

Some other lattices are discussed below −

Bounded Lattice

A lattice L becomes a bounded lattice if it has a greatest element 1 and a least element 0.

Complemented Lattice

A lattice L becomes a complemented lattice if it is a bounded lattice and if every element in the lattice has a complement. An element x has a complement x’ if $exists x(x land x’=0 and x lor x’ = 1)$

Distributive Lattice

If a lattice satisfies the following two distribute properties, it is called a distributive lattice.

$a lor (b land c) = (a lor b) land (a lor c)$

$a land (b lor c) = (a land b) lor (a land c)$

Modular Lattice

If a lattice satisfies the following property, it is called modular lattice.

$a land( b lor (a land d)) = (a land b) lor (a land d)$

Properties of Lattices

Idempotent Properties

$a lor a = a$

$a land a = a$

Absorption Properties

$a lor (a land b) = a$

$a land (a lor b) = a$

Commutative Properties

$a lor b = b lor a$

$a land b = b land a$

Associative Properties

$a lor (b lor c) = (a lor b) lor c$

$a land (b land c) = (a land b) land c$

Dual of a Lattice

The dual of a lattice is obtained by interchanging the $lor$ and $land$ operations.

Example

The dual of $lbrack a lor (b land c) brack is lbrack a land (b lor c) brack$