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Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is?
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Let H1 and H2 be any two distinct subgroup of a finite group G,each of...
Introduction:
In this problem, we are given two distinct subgroups H1 and H2 of a finite group G, each of order 2. We need to find the order of the smallest subgroup H that contains both H1 and H2.
Explanation:
To find the order of the smallest subgroup H, we can consider the group generated by H1 and H2. This group, denoted by, is the smallest subgroup containing both H1 and H2.
Proof:
Let's prove that is a subgroup of G.
1. Closure:
Since H1 and H2 are subgroups of G, they are closed under the group operation. Therefore, for any elements h1 and h2 in H1 and H2 respectively, their product h1h2 is also in H1 and H2. Hence, is closed under the group operation.
2. Identity:
Since H1 and H2 are subgroups of G, they contain the identity element e. Therefore, e is in both H1 and H2 and hence in. Therefore, contains the identity element.
3. Inverses:
Since H1 and H2 are subgroups of G, they contain the inverses of their elements. Therefore, for any element h1 in H1 and h2 in H2, their inverses h1^(-1) and h2^(-1) are also in H1 and H2 respectively. Hence, contains the inverses of its elements.
Therefore, satisfies all the conditions of being a subgroup. Note that is the smallest subgroup containing both H1 and H2.
Order of H:
Now, let's find the order of. Since H1 and H2 are both of order 2, the number of elements in is at most 4. This is because any element in can be written as a product of elements from H1 and H2, and since both H1 and H2 have order 2, there are at most 2 choices for each element.
If has exactly 4 elements, then it is a subgroup of order 4. Otherwise, it has fewer elements. However, since is the smallest subgroup containing both H1 and H2, it must be the case that has exactly 4 elements.
Therefore, the order of H is 4.
In this problem, we are given two distinct subgroups H1 and H2 of a finite group G, each of order 2. We need to find the order of the smallest subgroup H that contains both H1 and H2.
Explanation:
To find the order of the smallest subgroup H, we can consider the group generated by H1 and H2. This group, denoted by
Proof:
Let's prove that
1. Closure:
Since H1 and H2 are subgroups of G, they are closed under the group operation. Therefore, for any elements h1 and h2 in H1 and H2 respectively, their product h1h2 is also in H1 and H2. Hence,
2. Identity:
Since H1 and H2 are subgroups of G, they contain the identity element e. Therefore, e is in both H1 and H2 and hence in
3. Inverses:
Since H1 and H2 are subgroups of G, they contain the inverses of their elements. Therefore, for any element h1 in H1 and h2 in H2, their inverses h1^(-1) and h2^(-1) are also in H1 and H2 respectively. Hence,
Therefore,
Order of H:
Now, let's find the order of
If
Therefore, the order of H is 4.
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Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is?
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Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is?.
Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H1 and H2 be any two distinct subgroup of a finite group G,each of order 2. Let H be the smallest subgroup containg H1 and H2 then the order of H is?.
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