##### Algebraic Structure | Semi group | Monoid | Group | Abelian group | Finite group | Order of an element of a finite group | Subgroups | Lagrange's Theorem | Cyclic group

Algebraic Structure: A nonempty set 'S' is called an algebraic structure w.r.t binary operation '⊗' if (a⊗b) ∈ S ∀ a,b∈ S.
Semi group: An algebraic structure (S, ⊗) is called a semi group if (a⊗b)⊗c = a⊗(b⊗c) ∀ a,b∈ S , i.e, ⊗ is associative on S.
Monoid: A semigroup (S, ⊗) is called a monoid if there exists an element e∈ S such that (a⊗e) = (e⊗a) = a ,∀ a∈ S. Here e is called identity element.
Group: A monoid (S, ⊗) with identity element e is called a group if for each element a∈ S, there exists an element b∈ S, such that (a⊗b) = (b⊗a) = e. Here b is called inverse of a, and is denoted by a-1.
Abelian group: A group (G, ⊗) is said to be abelian if (a⊗b) = (b⊗a) ∀ a,b∈ G, i.e, ⊗ is commutative on G.
Finite group: A group with finite number of elements is called finite group.
Order of an element of a finite group: If G is a group and 'a'∈ G, then order of an element 'a' is the smallest positive integer such that an is identity element. Order of a group is represented by O(G).
Subgroups: Let (G,⊗) be a group. A subset 'H' of 'G' is called a subgraph of 'G' if (H,⊗) is a group.
Lagrange's Theorem: If 'H' is a subgroup of finite group (G,⊗), then O(H) is the divisor of O(G). The converse in need not to be true.
Cyclic group: A group (G,⊗) is called a cyclic group if there exists an element 'a'∈G such that every element of 'G' can be written as an for some integer n. Here 'a' is called generator
Note:
Let (G,⊗) be a cyclic group of order 'n' with generator 'a' then
I. The number of generator in G = ∅(n) = Euler function of n.
II. am is also a generator of G if GCD(m,n) =1.