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Let H and N be subgroup and normal subgroup of a group G respectively then
- a)N is normal subgroup of HN.
- b)N is normal subgroup of HN/G.
- c)N is cyclic subgroup.
- d)N is quotient subgroup.
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let H and N be subgroup and normal subgroup of a group G respectively ...
Suppose e∈ H and for any arbitrary x ∈ N.
Now HN is a subgroup of G, N is a subgroup of G and N ≤ HN therefore N is a subgroup of HN let hx be an arbitary element of HN st.
again x1 be any arbitary element of N then.
..(1)Since N Δ G therefore
Most Upvoted Answer
Let H and N be subgroup and normal subgroup of a group G respectively ...
Explanation:
To prove that option 'A' is correct, we need to show that N is a normal subgroup of HN.
Definition of a normal subgroup:
A subgroup N of a group G is called a normal subgroup if and only if for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Proof:
We need to show that for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Let's take an arbitrary element nh in HN, where n is an element of N and h is an element of H.
Now, let's consider the conjugate of nh by an element h' in H: (h')^(-1)(nh)(h').
Since N is a normal subgroup of G, we know that for every element n' in N and every element h'' in H, the product h''n'h''^(-1) is also in N.
Therefore, in our case, (h')^(-1)(nh)(h') is also in N.
Now, let's consider the product (h')^(-1)(nh)(h')n^(-1).
Since N is a subgroup of G, it is closed under the group operation. Therefore, the product nh and n^(-1) is also in N.
So, (h')^(-1)(nh)(h')n^(-1) is in N.
Now, let's consider the product h'(h')^(-1)(nh)(h')n^(-1).
Since G is a group, it is closed under the group operation. Therefore, the product h'(h')^(-1) is the identity element e.
So, h'(h')^(-1)(nh)(h')n^(-1) = enh'n^(-1) is in N.
Therefore, we have shown that for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Hence, option 'A' is correct.
To prove that option 'A' is correct, we need to show that N is a normal subgroup of HN.
Definition of a normal subgroup:
A subgroup N of a group G is called a normal subgroup if and only if for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Proof:
We need to show that for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Let's take an arbitrary element nh in HN, where n is an element of N and h is an element of H.
Now, let's consider the conjugate of nh by an element h' in H: (h')^(-1)(nh)(h').
Since N is a normal subgroup of G, we know that for every element n' in N and every element h'' in H, the product h''n'h''^(-1) is also in N.
Therefore, in our case, (h')^(-1)(nh)(h') is also in N.
Now, let's consider the product (h')^(-1)(nh)(h')n^(-1).
Since N is a subgroup of G, it is closed under the group operation. Therefore, the product nh and n^(-1) is also in N.
So, (h')^(-1)(nh)(h')n^(-1) is in N.
Now, let's consider the product h'(h')^(-1)(nh)(h')n^(-1).
Since G is a group, it is closed under the group operation. Therefore, the product h'(h')^(-1) is the identity element e.
So, h'(h')^(-1)(nh)(h')n^(-1) = enh'n^(-1) is in N.
Therefore, we have shown that for every element h in H and every element n in N, the product hnh^(-1) is also in N.
Hence, option 'A' is correct.
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Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. Can you explain this answer?
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Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. 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 Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. Can you explain this answer?.
Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. 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 Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H and N be subgroup and normal subgroup of a group G respectively thena)N is normal subgroup of HN.b)N is normal subgroup of HN/G.c)N is cyclic subgroup.d)N is quotient subgroup.Correct answer is option 'A'. Can you explain this answer?.
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