Groups

Group: A non empty set of elements, G is said to form a group if in G there is defined a binary operator * called product such that
          (a) a, b ∈ G ⇒ a * b ∈ G.                                                                      (Closed)
          (b) a, b, c ∈ G ⇒ a * (b * c) = (a * b) * c                                            (Associative)
          (c) ∃ e ∈ G : a * e = e * a = a, for all a ∈ G.                                       (Existence of identity)
          (d) For every a ∈ G, ∃ a^-1 ∈ G : a * a^-1= a^-1 * a = e        (Existence of inverse)

Finite and Infinite Group: If the set G in the group <G, *> is a finite set, then it is called a finite group otherwise it is called an infinite group.

Abelian group: A group G is Abelian (Commutative) if for every a, b ∈ G ⇒ a * b = b * a.
       If G is a non-empty set and * is any binary operation defined on G, then (G, *) is -
           (a) Quasi-group: a, b ∈ G ⇒ a * b ∈ G. 
           (b) Semi-group: a, b ∈ G 
            ⇒ a * b ∈ G and a, b, c ∈ G ⇒ a * (b * c) = (a * b) * c .
           (c) Monoid: a, b ∈ G ⇒ a . b ∈ G, a, b, c ∈ G
            ⇒ a * (b * c) = (a * b) * c
              ∃ e ∈ G : a * e = e * a = a.
             i.e., semi-group is a quasi-group with associativity.
             Monoid is a semi-group with identity.
            Abelian group is a group with commutativity.

Order of a group: The number of distinct elements in G is called an Order of a group. and is denoted by o(G) or |G|.
If a group G has n elements, then o(G) = n.
 Note: The order of an infinite group is not defined or we can say that the order is infinite.

                                      Subgroup

A non-empty subset H of G, is a subgroup of group G, if H is a group on the operator of G.

Right and left cosets: H is a subgroup of group G, a ∈ G, then
          Right coset of H in G is Ha = {ha : h ∈ H}
          Left coset of H in G is aH = {ah : h ∈ H}

Index of subgroup:  If H is a subgroup of G, the index of H in G is the number of distinct right cosets of H in G.

Order (period) of elements: If G is a group, a ∈ G, the order of a (period of a), is the least positive integer 
m : a^m = e, o (a) = m.

Product subgroups: HK = {x ∈ G : x = hk ; h ∈ H, k ∈ K}, H, G are subgroups of G.

Lemma: If H is a non-empty finite subset of G and H is closed the H is a subgroup of G.

Lemma: ∀ a ∈ G
                  Ha = {x ∈ G : a ≡ x mod H}.

Lemma:  There is one-to-one correspondence between to right cosets of H in G.

Normal subgroup: A subgroup N of G is normal subgroup if ∀ g ∈ G and n ∈ N, 
gng^-1∈N.

Quotient group: If G is a group, N is normal subgroup of G, then group G/N is called quotient (factor) group.

Lemma: N is normal subgroup of G iff gNg^-1 = N, for g ∈ G.

Lemma: N is a normal subgroup of G iff Na = aN, ∀ a ∈ G.

Lemma: N is a normal subgroup of G iff (Na)(Nb) = Nab.

Lemma: N is a normal subgroup of G, G is finite group then 
                           o(G/N) = o(G)/o(N).

Homomorphism: A mapping ∅ from group G into a group G̅ is said to be a homomorphism if ∀ a, b ∈ G, ∅ (ab) = ∅ (a) ∅ (b).

Kernel: If ∅ is a homomorphism of G into  G̅, the kernel of ∅, K_∅ is defined by  

K_∅ = {x ∈ G : ∅ = e̅ ,e̅ an identity element in  G̅}.

Isomorphism: A homomorphism ∅ from G into G̅ is an isomorphism if ∅ is one-to-one.

Isomorphic: Two groups G and G* are isomorphic if there is an isomorphism of G onto G*,

(G ≈ G*).

Automorphism: A homomorphism of a group G onto itself is called Automorphism.