In mathematics, groups are structures that represent sets of elements combined according to certain rules. A group consists of a set (G) and an operation (∘) used to combine the elements of that set. For instance, this operation can be addition (+), multiplication (×), or exclusive OR (⊕).

To qualify as a group, a set must satisfy the following four conditions:

1. Closure: The combination of any two elements in the set under the operation must result in an element that is also in the set. For example, the set of integers (Z) is closed under addition, as the sum of any two integers is also an integer.
2. Associativity: The order in which the operation is performed should not affect the result. This means (a∘b)∘c must hold for all elements a, b, and c in the set.
3. Identity Element: There must be an element in the set such that when it is combined with any other element under the operation, it does not change the other element. For example, in the case of addition, the identity element is 0, as a + 0 = a.
4. Inverse Element: Every element in the set must have an inverse such that when combined with the original element under the operation, the result is the identity element. For example, the additive inverse of a is −a, because a + (−a) = 0.

These properties are essential for the set and operation to form a group, providing a foundation for more advanced structures in algebra and applications in various mathematical fields.

3.1. Commutative (Abelian) Group

If the operation within a group does not depend on the order of the elements, this group is called a commutative or Abelian group. For instance, the set of integers with addition is an Abelian group because a+b=b+a

32. Subgroup

A subset HH of a group (G,∘) is called a subgroup if it satisfies the group axioms under the same operation as G. This is denoted as H≤G. In other words, if H maintains the group properties within itself, it indicates that H is part of G.

Let’s discuss this with an example:

• Consider the set of integers ZZ with the operation of addition (+). We can investigate whether the set of positive integers {1,2,3,4,…}{1,2,3,4,…} can be a subgroup.

1. Inverse Element Condition: This condition is not satisfied. For example, the additive inverse of 3 is -3, and -3 is not part of the set of positive integers.

This example shows that the set of positive integers is not a subgroup because it fails to satisfy the inverse element condition.

Now, what if we consider the set of even integers instead?

• The sum of any two even integers is also an even integer (e.g., 2+4=6).
• The identity element for addition, 0, is included in this set.
• The additive inverse of any even integer is also an even integer (e.g., the inverse of 4 is -4).

Therefore, the set of even integers is a subgroup of the set of integers under the operation of addition.