# Group

**GROUP DEFINITION**: The non-empty set A together with a binary operation, denoted by the just a position α1α2 for α1, α2 in A, constitutes a group whenever the following four axioms are satisfied

Axiom (i) (closure) : For all α1, α2 in A, α1α2 is also an element of A.

Axiom (ii) (associativity) : For all α1, α2, α3 in A, α1(α2α3) = (α1α2) α3

Axiom (iii) (identity) : There is an element i in A such that iα = αi = α for all α in A

Axiom (iv) (inversion) : If axiom (iii) holds, then for each α in A, there is an element denoted

α^{– 1} such that αα^{– 1} = α^{– 1}α = i.

**PERMUTATION**: A one-one mapping from a finite set onto itself is called a permutation.

**PERMUTATION GROUP**: The usual composition of mappings provides a binary operation for permutations on the same set. Whenever a collection of permutations is closed with respect to this composition, Axioms (ii), (iii) and (iv) are automatically satisfied and it is called a permutation group.

**Note :** If a permutation group A acts on object set X then | A | is the order of this group and | X | is the degree.

**ISOMORPHIC GROUPS**: If A and B are permutation groups acting on the sets X and Y then A ≅ B, means that A and B are isomorphic groups.

Here A ≡ B indicates not only isomorphism but that A and B are identical permutation groups.

If there is one-one map h : A ↔ B between the permutations such that for all α1, α2 in A.

h(α1 α2) = h(α1) h(α2) (i.e., A ≅ B)

Also, if there is a one-one map f : X ↔ Y between the objects such that for all x in X and α in A,

f(αx) = h(α) f(x) (i.e., A ≡ B).

**AUTOMORPHISM OF A GROUP**: An automorphism of a group G is an isomorphism of G with itself. Thus each automorphism α of G is a permutation of the point set V which preserves adjacency. Of course, α sends any point onto another of the same degree. Obviously any automorphism followed by another is also an automorphism, hence the automorphisms of G form a permutation group Γ(G).

**LINE-GROUP**: The point-group of G induces another permutation group Γ1(G), called the line-group of G which acts on the liens of G.

**OPERATIONS ON PERMUTATION GROUPS**

**Sum group**: Let A be a permutation group of order m = | A | and degree d acting on the set X = {x1, x2, ..... xd} and let B be another permutation group of order n = | B | and degree e actng on the set Y = {y1, y2, ..... ye}. The sum A B is a permutation group which acts on the disjoint union X ∪ Y and whose elements are all the ordered pairs of permutations α in A and β in B, written α β. Any object Z of X ∪ Y is permuted by α β according to the rule

**Product Group**: The product A × B of A and B is a permutation group which acts on the set X × Y and whose permutations are all the ordered pairs written α × β of permutations α in A and β in B. The object (x, y) of X × Y is permuted by α × β as expected (α × β)(x, y) = (αx, βy).

**Composition Group**: The composition A[B] of ‘A around B’ also acts on X × Y. For each α in A and any sequence (β1, β2, ..... βd) of d permutations in B, there is a unique permutation in A[B] written (α ; β1, β2, ..... βd) such that for (xi, yi) in X × Y.

**Power Group**: The power group denoted by BA acts on YX, the set of all functions from X into Y. Assume that the power group acts on more than one function. For each pair of permutations α in A and β in B there is a unique permutation, written βα in BA. The action of βα on any function f in YX by the following equation which gives the image of each x ∈ X under the function βα f : (βαf)(x) = βf(αx).