What are the groups?
Groups are the branch of abstract algebra.
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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 É‘ be a generator of G.Then o(É‘)=8, so we can write
G={É‘,É‘^2,É‘^3,É‘^4,É‘^5,É‘^6,É‘^7,É‘^8}.
7is prime to 8, therefore É‘^7 is also a generator of G.
5 is prime to 8, therefore É‘^5 is also a generator of G.
3 is prime to 8, therefore É‘^3 is also a generator of G.
So there are only four generators of G i.e.,É‘,É‘^3,É‘^5,É‘.
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