Groups in mathematics

What are the groups?

Groups are the branch of abstract algebra.

How does gruups help?

It has helped in developing chemistry, physics, and computer science.

we define groups

1.Binary Operation

Let S be a non-empty set.Any function*:SxS→S is called binary operation on S.

Example-A binary operation *  on S and (a,b)⋿S xS,we denote *(a,b) by a*b
Now* binary operation on a set S.We say that
1.* is closed on a subset T of S if a*b⋿ T ∀ a,b ⋿ T
2. * is associative if for all a,b,c ⋿ S,(a*b)*c =a*(b*c)
3.* is commutative if for all a,b ⋿ S, a*b=b*a

What are semi Group?

•An algebraic structure(G,*) is called a semigroup if the binary operation* is associative in G,if(a*b)*c=a*(b*c)  all a,b,cɛG.

•The set of N of all-natural number is a semigroup with respect to the operation of addition of natural numbers. Addition is an associative operation on N.

•The algebraic structures(N,*),(I,+) and(R,+) are also semigroup.

How many generators are of the cyclic group G of order 8?

Let ɑ be a generator of G.Then o(ɑ)=8, so we can write

G={ɑ,ɑ^2,ɑ^3,ɑ^4,ɑ^5,ɑ^6,ɑ^7,ɑ^8}.

7is prime to 8, therefore ɑ^7 is also a generator of G.

5 is prime to 8, therefore ɑ^5 is also a generator of G.

3 is prime to 8, therefore ɑ^3 is also a generator of G.

So there are only four generators of G i.e.,ɑ,ɑ^3,ɑ^5,ɑ.