Mathematical Foundation of Computer Science

DHARM

86 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

×1ωω^2

11 ωω^2

ωωω^21

ω^2 ω^21 ω

Fig 4.5 Operation Table.

4.8 Subgroups..........................................................................................................................

Let algebraic system (X, ) is a group, where X be a nonempty set and be a binary operation
on X. A subset Y of X such that (Y, ) is said to be a subgroup of X if (Y, ) is also a group
together following conditions are satisfie,

1. is closed,


2. is associative,


3. Existence of an identity element ê ∈ Y, where ê is the identity element of (X, ), and


4. Existence of unique an inverse for (X, ) and also for (Y, ).
   We can use the following notation for a subgroup,
   Y < X, where (Y, ) is the subgroup of (X. ).
   For example, algebraic system (I, +), where I is the set of integers and operation + is
   usual addition operations of integers, is a group. Also, I+ ⊆ I (where I+ is the set of all positive
   integers) and since, (I+, +) is a group and satisfy all the restrictions 1 – 4 therefore, (I+, +) is a
   subgroup.
   We further define, if Y and Z are two subsets of a group X, then
   l YZ = (yz/y ∈ Y, z ∈ Z} and
   l Y–1 = {y–1/y ∈ Y}; inverse of Y is also the subset of group X.
   l We define, Yk (for k = 0, 1, 2, ....) i.e.
   Y^1 = Y; Y^2 = YY; ........; similarly Yk+1 = Yk Y; and Y^0 = {ê}
   where,
   Y^2 = {y 1 y 2 /y 1 , y 2 ∈ Y}, and so on.
   The subset Y–k is defined as (Y–1)k.
   In particular, we write, for x ∈ X,
   Y x = {yx/y ∈ Y} or,
   x Y = {xy/y ∈ Y}
   If Y ≠ Ø, then we can see that
   Y X = X Y = X; X–1 = X; X ê = ê X = ê;
   Example 4.7. Let Y is a subgroup of X, i.e. Y < X if and only if YY–1 ⊂ Y.
   Sol. YY–1 ⊂ Y ⇒ if a, b ∈ Y then ab–1 ∈ Y.
   Necessity is obvious.
   For Sufficiency,
   if a ∈ Y ⇒ aa–1 ∈ YY–1 ⊂ Y;
   ⇒ aa–1 ∈ Y, i.e. ê ∈ Y.