GROUP THEORY Groupoid Semi Group Monoid group Abelian

GROUP THEORY Groupoid , Semi Group, Monoid, group, Abelian Group, order of a group, properties of a Group , Examples, Order of an element of a group, Modulo System, Permutation group.

Groupoid •

Monoid •

Group •

Abelian Group •

Order of a group •

Order of an Element of a Group • Let G be a group under multiplication. Let e be the identity element in G Suppose, a is any element of g, then the least positive integer n, if exist, such that an = e, is said to be order of an element a € G, and can be written as • o ( a) = n • In case, such a positive integer n does not exist, we say that the element a is of infinite or zero order • ii. Consider the additive group • Z= {……, -3, -2, -1, 0, 1, 2, 3, …. . } • 1. 0 = order of zero is one (finite) • but na≠ 0 is infinite.

Properties of Groups • • • Let (G, *) be a group, then The identity element ‘e’ is unique. There exists unique inverse in G. i. e. , a-1 G, A a*c=b*c⇒ a=b ( Right cancellation Law) c* a=c*b⇒ a=b ( Left cancellation Law) Where, a, b, c ∊G The left identity is also right identity i. e. , e. a=a and a. e=a The left inverse of an element is also its right inverse i. e. , a-1 a=e and aa-1 =e The equation a*x=b and y*a=b, where a, b∊ G have unique solutions in G, which are x=a-1 * b G and y=b * a -1 ∊G, respectively.

Permutation Groups • A one-one mapping f of a finite non-empty set S onto itself is called a permutation. • If the set Ѕ consists of n distinct elements, then a one-one mapping of Ѕ onto itself is called a permutation of degree n. • Let Ѕ = {a 1, a 2, …, an}. • Then, we denote a permutation f on the set S in a two-rowed notation. • • So that in the first row all the elements of S are written in certain order and • f(a 1)=b 1, f(a 2)=b 2……, f(a 1)=b 1, …. , f(an)=bn, • Where sare s.

Types of Permutations Equality of Permutations • Two permutations f and g of a set S are said to be equal, if • f(a) = g(a), Ɐ a ∊ S. • e. g. , if • And • Of degree 3, then we have • f=g since, f(1)= g(1)=3, f(2)=g(2)=1 f(3)=g(3)=2

Identity Permutation • A permutation on the set S is called the identity permutation, it maps each element of S onto itself, • It is usually denoted by the symbol l. • Thus, l(a)=a, Ɐ a ∊ S • e. g, • is the identity permutation of degree n.

Inverse Permutations • Since, a permutation is one-one onto mapping and hence, it is invertible, i. e. , every permutation f on a set P= {a 1, a 2, …. . an} has a unique inverse permutation denoted by f -1 • e. g. , If • and

Permutation Group (Symmetric Group) • Let A (S) denote the set of all permutation on a non-empty set S. • Then, A(S) form a group under the composition of maps. Moreover, if S contains n elements, then the permutation group A(S) contains n! elements. The group of n! permutations of a set n elements is called symmetric group ( Sn) of degree n. • Example 6. Write down all the elements of the permutation group ( or symmetric group) S 3 on three elements 1, 2 and 3. • Solution. Let S= {1, 2, 3} • Then, there are 3! = 6 elements in S 3. • • • which can be written as