Group Theory Part-1 With QNA(Theoritical Approach)

Algebraic Structure:

A non-empty set equipped one or more binary operation.   

  Ex: ( G,*) , ( S,*) etc

NOTE: Here G and S are non-empty sets and  ' * ' is called binary operations.

Group:

A group is an algebraic structure ' G ' equipped with one binary operation ' * '

i.e; (G,*) along with the following four properties.

1. Closure Property:

For every a,b ∈ G ⇒ a *b ∈ G

2.Associative Property:

For every a,b,c ∈ G ⇒ a*(b*c) =(a*b)*c

3.Identity Property:

There is an element e∈G ,such that a*e=a=e*a

Here ' e ' is the identity element in G.

4.Inverse Property:

For every a∈G , There exist an element  a^-1 ∈ G such that a*a^-1 =e =a^-1*a

Here a^-1 is known as the inverse element of a.

Abelian Groups:

A group is called an abelian group if it satisfies the commutative property.

i.e; For every a,b∈ G ⇒ a*b=b*a

Other Algebraic Structures:

Groupoid:

If any algebraic structure which equipped only one binary structure satisfies the closure property only. This is called as Groupoid.

Semi Group:

If any algebraic structure which equipped only one binary structure satisfies two property (closure and associative) only is called Semi Group.

Monoid:

If any algebraic structure which equipped only one binary structure satisfies Closure,associative and identity laws is called a monoid.

Monoids are semigroups with identity.

SubGroup:

It is a subset of a group which is itself a group and satisfy all the properties of a group.

It should form the group under the same operation of its superset.

Cyclic Group:

A group G is said to be a cyclic group if there exist an element a ∈ G  such that every element of G can be expressed as some power of a.

          If G is a group generated by 'a' we say that 'a ' is a generator of G.

     G=(a)

Note:
A cyclic group is always an abelian group but every abelian group is not a cyclic group. For instance, the rational numbers under addition is an abelian group but is not a cyclic one.

I am Posting some Easy Question below. Try to solve.(Good Luck)

IF any Doubt arise just Comment us.

1. A non empty set A is termed as an algebraic structure ________
a) with respect to binary operation *
b) with respect to ternary operation ?
c) with respect to binary operation +
d) with respect to unary operation –

2. An algebraic structure _________ is called a semigroup.
a) (P, *)
b) (Q, +, *)
c) (P, +)
d) (+, *)

3. Condition for monoid is __________
a) (a+e)=a
b) (a*e)=(a+e)
c) a=(a*(a+e)
d) (a*e)=(e*a)=a

4. A cyclic group can be generated by a/an ________ element.
a) singular
b) non-singular
c) inverse
d) multiplicative

5. How many properties can be held by a group?
a) 2
b) 3
c) 5
d) 4

6.A group (M,*) is said to be abelian if ___________
a) (x+y)=(y+x)
b) (x*y)=(y*x)
c) (x+y)=x
d) (y*x)=(x+y)

7. A monoid is called a group if _______
a) (a*a)=a=(a+c)
b) (a*c)=(a+c)
c) (a+c)=a
d) (a*c)=(c*a)=e

8.A cyclic group is always _________
a) abelian group
b) monoid
c) semigroup
d) subgroup

Answers:

1.a,2.a,3.d,4.a,5.c,6.b,7.d,8.a