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Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K?
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Let G be a group of order 2022 .let H and K be subgroup , order of H i...
Question:
Let G be a group of order 2022. Let H and K be subgroups, where the order of H is 337 and the order of K is 674. If H ∪ K is a subgroup, then:
1. What is the order of H ∪ K (if possible)?
2. Is H a normal subgroup of H ∪ K?
3. Is K a normal subgroup of H ∪ K?
Answer:
In order to answer the questions, we need to understand the concepts of subgroups and normal subgroups.
Subgroup:
A subgroup is a subset of a group that is itself a group under the same operation. In this case, we have two subgroups H and K.
Normal Subgroup:
A subgroup H of a group G is said to be a normal subgroup if and only if for every element 'g' in G, the conjugate of H by g is contained in H. In other words, gHg^(-1) = H for all g in G.
1. Order of H ∪ K:
To find the order of H ∪ K, we need to find the smallest possible number that divides both 337 and 674. The order of H ∪ K is equal to the least common multiple (LCM) of the orders of H and K.
The prime factorization of 337 is 337 = 1 * 337.
The prime factorization of 674 is 674 = 2 * 337.
The LCM of 337 and 674 is 2 * 337 = 674.
Therefore, the order of H ∪ K is 674.
2. Is H a normal subgroup of H ∪ K?
To determine if H is a normal subgroup of H ∪ K, we need to check if every element of H conjugated by any element of H ∪ K is contained in H.
Since H ∪ K is a subgroup, it must contain the identity element e. For any element h in H, we have eh = he = h, which is an element of H. Therefore, H is closed under conjugation by elements of H ∪ K.
Thus, H is a normal subgroup of H ∪ K.
3. Is K a normal subgroup of H ∪ K?
Similarly, to determine if K is a normal subgroup of H ∪ K, we need to check if every element of K conjugated by any element of H ∪ K is contained in K.
Since K is a subgroup of G and H ∪ K is a subgroup, H ∪ K contains all the elements of K. Therefore, conjugating any element of K by any element of H ∪ K will result in an element that is still in K.
Thus, K is also a normal subgroup of H ∪ K.
In summary:
1. The order of H ∪ K is 674.
2. H is a normal subgroup of H ∪ K.
3. K is also a normal subgroup of H ∪ K.
Let G be a group of order 2022. Let H and K be subgroups, where the order of H is 337 and the order of K is 674. If H ∪ K is a subgroup, then:
1. What is the order of H ∪ K (if possible)?
2. Is H a normal subgroup of H ∪ K?
3. Is K a normal subgroup of H ∪ K?
Answer:
In order to answer the questions, we need to understand the concepts of subgroups and normal subgroups.
Subgroup:
A subgroup is a subset of a group that is itself a group under the same operation. In this case, we have two subgroups H and K.
Normal Subgroup:
A subgroup H of a group G is said to be a normal subgroup if and only if for every element 'g' in G, the conjugate of H by g is contained in H. In other words, gHg^(-1) = H for all g in G.
1. Order of H ∪ K:
To find the order of H ∪ K, we need to find the smallest possible number that divides both 337 and 674. The order of H ∪ K is equal to the least common multiple (LCM) of the orders of H and K.
The prime factorization of 337 is 337 = 1 * 337.
The prime factorization of 674 is 674 = 2 * 337.
The LCM of 337 and 674 is 2 * 337 = 674.
Therefore, the order of H ∪ K is 674.
2. Is H a normal subgroup of H ∪ K?
To determine if H is a normal subgroup of H ∪ K, we need to check if every element of H conjugated by any element of H ∪ K is contained in H.
Since H ∪ K is a subgroup, it must contain the identity element e. For any element h in H, we have eh = he = h, which is an element of H. Therefore, H is closed under conjugation by elements of H ∪ K.
Thus, H is a normal subgroup of H ∪ K.
3. Is K a normal subgroup of H ∪ K?
Similarly, to determine if K is a normal subgroup of H ∪ K, we need to check if every element of K conjugated by any element of H ∪ K is contained in K.
Since K is a subgroup of G and H ∪ K is a subgroup, H ∪ K contains all the elements of K. Therefore, conjugating any element of K by any element of H ∪ K will result in an element that is still in K.
Thus, K is also a normal subgroup of H ∪ K.
In summary:
1. The order of H ∪ K is 674.
2. H is a normal subgroup of H ∪ K.
3. K is also a normal subgroup of H ∪ K.
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Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K?
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Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K?.
Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K?.
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Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K?, a detailed solution for Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? has been provided alongside types of Let G be a group of order 2022 .let H and K be subgroup , order of H is 337 and order of k is 674 .if H union K is subgroup then 1. What is the order of H union K (if possible)? 2. Is H is a normal subgroup of H union K 3. Is K is a normal subgroup of H union K? theory, EduRev gives you an
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