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If a group has 5 element of order 2,then if it is abelian then is it possible?
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If a group has 5 element of order 2,then if it is abelian then is it p...
Introduction:
In this question, we are given a group that has 5 elements of order 2. We need to determine whether this group can be abelian or not.
Understanding the problem:
To solve this problem, we need to understand the concept of order and abelian groups.
Order of an element:
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In other words, the order of an element is the number of times we need to apply the group operation to get the identity element.
Abelian groups:
An abelian group, also known as a commutative group, is a group in which the group operation is commutative. In other words, for any two elements a and b in the group, the result of the operation ab is the same as the result of the operation ba.
Analysis:
Let's assume that our group G has 5 elements of order 2, denoted by a, b, c, d, and e. Since the order of each element is 2, we have a^2 = b^2 = c^2 = d^2 = e^2 = e (identity element).
If G is an abelian group, then for any two elements a and b in G, ab = ba. Let's consider two arbitrary elements a and b in G.
Proof:
If a and b are distinct elements, then ab and ba must be distinct as well. This is because if ab = ba, then multiplying both sides by a^-1 on the left gives b = a^-1ba = a^-1ab = b, which contradicts the assumption that a and b are distinct.
Now, let's consider the product ab. Since a and b are distinct elements, ab cannot be equal to a or b because the order of a and b is 2. Similarly, the product ba cannot be equal to a or b. Therefore, ab and ba must be distinct elements in the group.
However, we have assumed that our group has 5 elements of order 2. This means that there are only 5 distinct elements in the group. Since ab and ba are distinct, this implies that our group must have at least 6 distinct elements, which contradicts the given information.
Conclusion:
Therefore, it is not possible for a group with 5 elements of order 2 to be abelian.
In this question, we are given a group that has 5 elements of order 2. We need to determine whether this group can be abelian or not.
Understanding the problem:
To solve this problem, we need to understand the concept of order and abelian groups.
Order of an element:
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In other words, the order of an element is the number of times we need to apply the group operation to get the identity element.
Abelian groups:
An abelian group, also known as a commutative group, is a group in which the group operation is commutative. In other words, for any two elements a and b in the group, the result of the operation ab is the same as the result of the operation ba.
Analysis:
Let's assume that our group G has 5 elements of order 2, denoted by a, b, c, d, and e. Since the order of each element is 2, we have a^2 = b^2 = c^2 = d^2 = e^2 = e (identity element).
If G is an abelian group, then for any two elements a and b in G, ab = ba. Let's consider two arbitrary elements a and b in G.
Proof:
If a and b are distinct elements, then ab and ba must be distinct as well. This is because if ab = ba, then multiplying both sides by a^-1 on the left gives b = a^-1ba = a^-1ab = b, which contradicts the assumption that a and b are distinct.
Now, let's consider the product ab. Since a and b are distinct elements, ab cannot be equal to a or b because the order of a and b is 2. Similarly, the product ba cannot be equal to a or b. Therefore, ab and ba must be distinct elements in the group.
However, we have assumed that our group has 5 elements of order 2. This means that there are only 5 distinct elements in the group. Since ab and ba are distinct, this implies that our group must have at least 6 distinct elements, which contradicts the given information.
Conclusion:
Therefore, it is not possible for a group with 5 elements of order 2 to be abelian.
Community Answer
If a group has 5 element of order 2,then if it is abelian then is it p...
Introduction:
In this question, we are given a group that has 5 elements of order 2, and we need to determine if it is possible for this group to be abelian. To answer this question, we will first define what it means for a group to be abelian and then explore the properties of a group with 5 elements of order 2.
Abelian Groups:
An abelian group, also known as a commutative group, is a group where the order of the elements does not matter. In other words, for any two elements a and b in the group, ab = ba. This property is named after the mathematician Niels Henrik Abel.
Properties of a Group with 5 Elements of Order 2:
Let's assume that our group G has 5 elements of order 2 - let's call them a, b, c, d, and e. Since the order of an element is the smallest positive integer n such that a^n = e (the identity element), we have the following properties for our group:
- a^2 = e
- b^2 = e
- c^2 = e
- d^2 = e
- e^2 = e
Proof by Contradiction:
To determine if our group is abelian, we need to verify if ab = ba for all pairs of elements a and b in the group. We will use a proof by contradiction to show that if our group has 5 elements of order 2, it cannot be abelian.
Assume that our group G is abelian, and let's consider the pair ab. Since a and b both have order 2, we have the following possibilities:
- If a = b, then ab = aa = a^2 = e.
- If a ≠ b, then ab ≠ ba, as this would contradict the abelian property.
Now, let's consider a pair of different elements, say a and b. If we assume that ab = ba, then we can write it as a multiplication table:
From the table, we can see that ab = ba only for the pair (a,a), which means that the only way for our group to be abelian is if a = b = c = d = e. However, this contradicts the given condition that the group has 5 elements of order 2.
Conclusion:
Therefore, it is
In this question, we are given a group that has 5 elements of order 2, and we need to determine if it is possible for this group to be abelian. To answer this question, we will first define what it means for a group to be abelian and then explore the properties of a group with 5 elements of order 2.
Abelian Groups:
An abelian group, also known as a commutative group, is a group where the order of the elements does not matter. In other words, for any two elements a and b in the group, ab = ba. This property is named after the mathematician Niels Henrik Abel.
Properties of a Group with 5 Elements of Order 2:
Let's assume that our group G has 5 elements of order 2 - let's call them a, b, c, d, and e. Since the order of an element is the smallest positive integer n such that a^n = e (the identity element), we have the following properties for our group:
- a^2 = e
- b^2 = e
- c^2 = e
- d^2 = e
- e^2 = e
Proof by Contradiction:
To determine if our group is abelian, we need to verify if ab = ba for all pairs of elements a and b in the group. We will use a proof by contradiction to show that if our group has 5 elements of order 2, it cannot be abelian.
Assume that our group G is abelian, and let's consider the pair ab. Since a and b both have order 2, we have the following possibilities:
- If a = b, then ab = aa = a^2 = e.
- If a ≠ b, then ab ≠ ba, as this would contradict the abelian property.
Now, let's consider a pair of different elements, say a and b. If we assume that ab = ba, then we can write it as a multiplication table:
| * | a | b | c | d | e |
|---|---|---|---|---|---|
| a | e | * | * | * | * |
| b | * | e | * | * | * |
| c | * | * | e | * | * |
| d | * | * | * | e | * |
| e | * | * | * | * | e |
From the table, we can see that ab = ba only for the pair (a,a), which means that the only way for our group to be abelian is if a = b = c = d = e. However, this contradicts the given condition that the group has 5 elements of order 2.
Conclusion:
Therefore, it is
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If a group has 5 element of order 2,then if it is abelian then is it possible?
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If a group has 5 element of order 2,then if it is abelian then is it possible? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If a group has 5 element of order 2,then if it is abelian then is it possible? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a group has 5 element of order 2,then if it is abelian then is it possible?.
If a group has 5 element of order 2,then if it is abelian then is it possible? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If a group has 5 element of order 2,then if it is abelian then is it possible? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a group has 5 element of order 2,then if it is abelian then is it possible?.
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