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Which of the following is TRUE for groups of even order?
- a)Such groups do not have a non-trivial proper subgroup
- b)There is no element which is the inverse o f itself
- c)The order of such groups is a power of 2
- d)There are atleast two elements whose inverses are the elements themselves
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Which of the following is TRUE for groups of even order?a)Such groups ...
If we take G = { 1 , - 1 , i, - i } then G has a non-trivial proper subgroup H = {1, - 1}
So, option (a) is discarded.
option (b) is discarded because identity is self inverse element in any group, option (c) is discarded because if we take Z6 = {0,1,2, 3,4, 5} then Z6 is a group but it is not power of 2.
For even order group apart from identity there will odd number of elements. Since inverse element exist in pairs so at least one element will be it own inverse including identity such elements will be two.
So, option (a) is discarded.
option (b) is discarded because identity is self inverse element in any group, option (c) is discarded because if we take Z6 = {0,1,2, 3,4, 5} then Z6 is a group but it is not power of 2.
For even order group apart from identity there will odd number of elements. Since inverse element exist in pairs so at least one element will be it own inverse including identity such elements will be two.
Most Upvoted Answer
Which of the following is TRUE for groups of even order?a)Such groups ...
Statement: There are at least two elements in a group of even order whose inverses are the elements themselves.
Explanation:
To prove that there are at least two elements in a group of even order whose inverses are the elements themselves, we can use the concept of the pigeonhole principle.
Pigeonhole Principle: If there are n+1 pigeons and n holes, then at least one hole must contain more than one pigeon.
Proof:
Let G be a group of even order, and let e be the identity element of G.
Consider the set S = {g ∈ G | g ≠ e and g = g^(-1)}, i.e., the set of elements in G which are not the identity and are their own inverses.
Now, let's consider the following cases:
Case 1: S contains at least two distinct elements.
In this case, we have at least two elements in G whose inverses are themselves, and the statement is true.
Case 2: S contains only one element.
In this case, we know that the identity element e is its own inverse, so we have found one element whose inverse is itself.
But since G has even order, i.e., the number of elements in G is even, and considering that S contains only one element, there must be at least one more element in G that is not in S. Let's call this element g.
Now, let's consider the element g^(-1). Since g is not in S, g^(-1) ≠ g, and since g is not its own inverse, g^(-1) ≠ g^(-1)^(-1) = g. Therefore, g^(-1) is a distinct element in G.
So, we have found two elements in G, g and g^(-1), such that their inverses are themselves.
Conclusion:
Therefore, it is proven that in a group of even order, there are at least two elements whose inverses are the elements themselves (option D is true).
Explanation:
To prove that there are at least two elements in a group of even order whose inverses are the elements themselves, we can use the concept of the pigeonhole principle.
Pigeonhole Principle: If there are n+1 pigeons and n holes, then at least one hole must contain more than one pigeon.
Proof:
Let G be a group of even order, and let e be the identity element of G.
Consider the set S = {g ∈ G | g ≠ e and g = g^(-1)}, i.e., the set of elements in G which are not the identity and are their own inverses.
Now, let's consider the following cases:
Case 1: S contains at least two distinct elements.
In this case, we have at least two elements in G whose inverses are themselves, and the statement is true.
Case 2: S contains only one element.
In this case, we know that the identity element e is its own inverse, so we have found one element whose inverse is itself.
But since G has even order, i.e., the number of elements in G is even, and considering that S contains only one element, there must be at least one more element in G that is not in S. Let's call this element g.
Now, let's consider the element g^(-1). Since g is not in S, g^(-1) ≠ g, and since g is not its own inverse, g^(-1) ≠ g^(-1)^(-1) = g. Therefore, g^(-1) is a distinct element in G.
So, we have found two elements in G, g and g^(-1), such that their inverses are themselves.
Conclusion:
Therefore, it is proven that in a group of even order, there are at least two elements whose inverses are the elements themselves (option D is true).
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Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer?
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Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer?.
Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is TRUE for groups of even order?a)Such groups do not have a non-trivial proper subgroupb)There is no element which is the inverse o f itselfc)The order of such groups is a power of 2d)There are atleast two elements whose inverses are the elements themselvesCorrect answer is option 'D'. Can you explain this answer?.
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