Question

Find all cyclic subgroups of quaternion group

Answer 1

We haven't had much practice with using the quaternion group at all. Although, we have been focusing more on the Klein-44 group but that's not the point. From what I understand, upon doing some internet browsing, that the quaternion group has order 88 since

Q={±1,±i,±j,±k}.
Q={±1,±i,±j,±k}.
Now, if I were to have a subgroup, call it HH, such that it has index 22 in, say, group GG, it would mean that half of the elements in GG lie in HH. So am I suppose to find all the subgroups of the quaternion group, then chose which ones have index 22? How would one begin to answer this question?

Answer 2

Note :

Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Cyclic group : A group G is called a cyclic group , if there exists an element a ∈ G , such that every element x ∈ G can be written as x = aⁿ for some integer n . And the element a is called the generator of G .

Subgroup : A non empty subset S of the group G is said to be a subgroup of G when it forms a group under the operation of G .

Answer :

Every proper sungroup of the quaternion group Q₈ is cyclic .

Explanation :

We have Q₈ = { ±1 , ±i , ±j , ±k }

ie. Order of Q₈ , o(Q₈) = 8

We know that , the order of a subgroup divides the order of its parent group . Thus , the possible orders of the subgroups of Q₈ are : 1 , 2 , 4 , 8 .

Now , the subgroups of Q₈ are :

H₁ = { 1 } = { e }

H₂ = { 1 , -1 }

H₃ = { 1 , -1 , i , -i }

H₄ = { 1 , -1 , j , -j }

H₅ = { 1 , -1 , k , -k }

H₆ = { ±1 , ±i , ±j , ±k } = Q₈

Here ,

H₁ , H₂ , H₃ , H₄ , H₅ are the proper sungroups of Q₈ .

Now ,

H₁ = { 1 } = < 1 >

H₂ = { 1 , -1 } = { (-1)² , (-1)¹ } = < -1 >

H₃ = { 1 , -1 , i , -i } = { i⁴ , i² , i¹ , i³ } = < i >

H₄ = { 1 , -1 , j , -j } = { j⁴ , j² , j¹ , j³ } = < j >

H₅ = { 1 , -1 , k , -k } = { k⁴ , k² , k¹ , k³ } = < k >

Clearly ,

Every proper sungroup of Q₈ are cyclic .