Introduction of subgroups in mathematics:

Subgroups in mathematics are the important branch of mathematics. A group is a algebraic structure that is the combination of element and operators. The resultant gives the third new element. For Example integers are the most common type of group under number theory. The properties are

* Closure

* Associativity

* Identity

* Inverse

The Important Properties Involving in Subgroups in Mathematics

They are,

Closure Property in mathematics:

Under addition:

For any two integer’s p and q, the sum p + q is also an integer. This property is known as closure property under addition.

Under Multiplication:

For all p, q in G, the resultant of the operation p * q is also in G. here G represents a group.

Example:

For all 2, 3 in N here N is the set of natural numbers, the result of the operation 2 * 3 that is 6 is also in the group of N.

This is one of the types of subgroup property.

Associativity property in mathematics:

Under Addition:

For all integers p, q and r, (p + q) + r = p + (q + r). This property is known as Associativity property under addition.

Under multiplication:

For all p, q and c in G, then the equation (p * q) * r = p * (q * r) property holds.

Example:

For all 5, 6 and 3in G, the equation (5 * 6) * 3 = 5* (6 * c3) holds. Here both left hand side and right hand side as the equal value 90.

This is one of the types of subgroup property.

Identity element in mathematics:

Under Addition:

If p is any integer, then 0 + p =p + 0 = p. Here 0 is known as the identity element

Under multiplication:

There exists an element e in G, such that for every element p in G, then the equation e * p = p* e = p property holds.

Example:

1.2 = 2.1 =2

Here 1 is the identity element.

This is one of the types of subgroup property.

Inverse Element in mathematics:

Under addition:

For every integer p, there is an integer q such that p + q = q+ p = 0. The integer q is known as inverse of p and it is denoted −p.

Under multiplication:

For every p in G, there exists an element q in G such that p * q = q* p= e, here e is known as the identity element.

Example:

-3 is the inverse element of 3

Because -3 + 3 = 3 – 3 = 0

Here zero is the identity element.

This is one of the types of subgroup property.

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