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VOOZH | about |
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VOOZH | about |
Groups are an important concept for many branches of mathematics, including geometry, algebra, and number theory. Clusters are formed as basic systems. A group is a group that has "the function of combining any two entities into a third entity." They provide an effective basis for investigating algebraic structures, symmetries, and permutations. A basic group is a collection of two actions that satisfy a set of requirements. To analyze more complex statistical data in depth, it is important to understand the basic characteristics of the categories.
Table of Content
A group is a set G combined with a binary operation ∗ (often called multiplication or addition depending on context) that satisfies the following four properties:
1. Closure: For every pair of elements, a,b in G, operation a b's result is also in G.
Example: Consider the set of integers Z with addition as the operation. For any two integers a and a+b are also an integer, hence Z is closed under addition.
2. Associativity: For every three elements a,b,c in G, the equation (a∗b)∗c=a∗(b∗c) holds.
Example: For any three integers a,b, and c in Z, (a+b) +c = a+ (b+ c)
3. Identity Element: There exists an element e in G such that for every element a in G, the equation e∗a=a∗e=a holds.
Example: In the group of integers Z with addition, the identity element is 0 because for any integer a, a+0=0+a=a
4. Inverse Element: For each element a in G, there exists an element b in G such that a∗b=b∗a=e, where e is the identity element.
Example: For any integer a in Z, the inverse element is −a because a+(−a)=(−a)+a=0.
If a , b, c ∈ G then, is a o b = a o c ⇒ b = c
Proof:
Given a o b = a o c, for every a, b, c ∈ G
Operating on the left with a-1, where a-1 ∈ G we have
a-1 o (a o b) = a-1 o (a o c)
or (a-1 o a) o b = (a-1 o a) o c [using associative property]
or e o b = e o c, [using inverse property]
or b = c, [using identity property]
Note that a o b is also written as ab.
This is known as the left cancellation law.
For every a ∈ G , e o a = a = a o e, where e is the identity element. i.e. The left identity element is also the right identity element.
Proof:
If a-1 be the left inverse of a, then
a-1 o (a o e) = (a-1 o a) o e [using associative property]
or a-1 o (a o e) = e o e [using inverse property]
= e [using identity property]
or a-1 o (a o e) = a-1 o a [using inverse property]
i.e. a-1 o (a o e) = a-1 o a
Hence, a o e = a by property-1 i.e. left cancellation law. thus we find that e is also the right identity element and so it is called only the identity element.
For every a ∈ G , a-1 o a = e = a o a-1 i.e. the left inverse of an element is also its right inverse.
Proof:
a-1 o (a o a-1) = (a-1 o a) o a-1 [using identity property]
= e o a-1 [using inverse property]
= a-1 o e [by property 2]
i.e. a-1 o (a o a-1)= a-1 o e
Hence, a o a-1 = e, by left cancellation law.
Thus, we find that the left inverse a-1 of element a is also its right inverse and so a-1 is called only the inverse of a.
If a , b, c ∈ G then, is b o a = c o a ⇒ b = c
Proof:
Given a o b = a o c, for every a, b, c ∈ G
Operating on the left with a-1, where a-1 ∈ G we have
(b o a) o a-1 = (c o a) o a-1
or b o (a-1 o a) = c o (a-1 o a) [using associative property]
or b o e = c o e, [using inverse property]
or b = c, [using identity property]This is known as right cancellation law.
For every a , b ∈ G we have (a o b)-1 = b-1 o a-1 i.e. The inverse of the product (or the composite) of two elements a, b of group G is the product (or composite) of the inverses of the two elements taken in the reverse order.
Proof:
Let a-1 and b-1 be the inverses of a and b.
Now,(a o b) o (b-1 o a-1) = a o (b o b-1) o a-1 [using associative property]
= a o e o a-1 [using inverse property]
= a o a-1 [using identity property]
= e [using inverse property]
(a o b) o (b-1 o a-1) = e
Similarly, (b-1 o a-1) o ( a o b)= e
Therefore, by the definition of inverse b-1 o a-1 is the inverse of a o b. i.e. (a o b)-1=b-1 o a-1
This is known as the reversal rule.
The group G is called Abelian Group (or commutative) if the action ∗ satisfies the commutative property, that is, a∗b=b∗a for all a,b in G. Abelian groups are important because they have order in analysis is easier, more And all the groups are abelian.
Example: The set Z of collected integers is an abelian group because a+b=b+a for any integers a and b.
A group G is not Abelian if G contains elements a,b such that a∗b =b∗a. Non-Abelian groups are more complex and arise in situations where the function is important.
Example: The three-element symmetric group of variables s3 is abelian because in general one variable followed by another does not produce the same result as in the reverse order
Finite-infinite set: Sets can also be classified based on the number of elements. A group is finite if it has an infinite number of elements, and infinite if it has an infinite number of elements.
Example (finite set): The set of integers is a finite set including modulo n, θ, and sum modulo n.
Example (infinite set): The set Z of all collected integers is an infinite set.
Example 1:Prove that the set of integers under addition is a group.
Solution:
Closure: For any two integers a and b, their sum a + b is also an integer.
Associativity: For any integers a, b, and c, (a + b) + c = a + (b + c).
Identity Element: The identity element is 0, as a + 0 = 0 + a = a.
Inverse Element: For any integer a, the inverse is -a because a + (-a) = (-a) + a = 0.
Example 2: Prove that the set of non-zero real numbers under multiplication is a group.
Solution:
Closure: For any two non-zero real numbers a and b, their product a * b is also a non-zero real number.
Associativity: For any non-zero real numbers a, b, and c, (a b) c = a (b c).
Identity Element: The identity element is 1, as a 1 = 1 a = a.
Inverse Element: For any non-zero real number a, the inverse is 1/a because a (1/a) = (1/a) a = 1.
Example 3: Prove that the set of 2x2 invertible matrices under multiplication is a group.
Solution:
Closure: The product of any two invertible 2x2 matrices is also an invertible 2x2 matrix.
Associativity: For any invertible 2x2 matrices A, B, and C, (A B) C = A (B C).
Identity Element: The identity element is the 2x2 identity matrix I, as A I = I A = A for any invertible matrix A.
Inverse Element: For any invertible 2x2 matrix A, its inverse A^-1 satisfies A A^-1 = A^-1 A = I.
Example 4: Verify that the set of integers modulo n under addition modulo n is a group.
Solution:
Closure: For any two integers a and b, their sum modulo n (a + b) mod n is also an integer modulo n.
Associativity: For any integers a, b, and c, ((a + b) mod n + c) mod n = (a + (b + c) mod n) mod n.
Identity Element: The identity element is 0, as (a + 0) mod n = (0 + a) mod n = a mod n.
Inverse Element: For any integer a, the inverse is n - a because (a + (n - a)) mod n = 0 mod n = 0.
Example 5: Prove that the set of non-zero complex numbers is a group under multiplication.
Solution:
Closure: For any two non-zero complex numbers a and b, their product a * b is also a non-zero complex number.
Associativity: For any non-zero complex numbers a, b, and c, (a b) c = a (b c).
Identity Element: The identity element is 1, as a 1 = 1 a = a.
Inverse Element: For any non-zero complex number a, the inverse is 1/a because a (1/a) = (1/a) a = 1.
1. Prove that the set of all complex numbers under addition forms a group.
2. Verify that the set of positive rational numbers under multiplication forms a group.
3. Show that the set of integers modulo 5 under multiplication is not a group.
4. Check if the 2×2 matrices with determinant 1 form a group under multiplication.
5. Show that the non-zero real numbers do not form a group under multiplication.
6. Verify that the set of all real numbers under addition forms an abelian group.
7. Show that the set of 3×3 matrices with determinant 1 forms a group under matrix multiplication.
8. Determine whether the set of odd integers under addition forms a group.
9. Check if the non-zero complex numbers form a group under multiplication.
10. Prove that the set of all permutations of three elements forms a non-Abelian group.
Also Read:
Symmetry, operations, and transformations unite under groups, math's cornerstone structures. These versatile tools shape number theory, geometry, and algebra. Mastering group properties unlocks advanced mathematical realms. From simple sets to complex theories, groups form the bedrock of analytical thinking. Their study cultivates a deep understanding, essential for navigating higher mathematics. Groups bridge concepts, revealing hidden patterns and connections across diverse fields.
