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?

Question

Answer

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.

Similar Mathematics Doubts

If H1 ,H2 are the distinct subgroups of a finite group G , each of order 2 .H is the smallest subgroup of G containing H1 and H2 .Then find the order of H .?

Let g br a finite abelian group . Let h1 and h2 be two distinct subgroups of G of index 3 each then the index of h1intersection h2 in G is?

Let G be a finite group of order 200. Then find total number of subgroup of G of order 25. Correct answer is '1'. Can you explain this answer?

Check the correct statement.

Let G be a group and H and K are subgroups sach H has a finite index in G then,?

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams