Let Q be the set of eight elements, {±1, ±i, ±j, ±k}. Find List all subgroups of

Question

Let Q be the set of eight elements, {±1, ±i, ±j, ±k}. Find List all subgroups of Q, including the trivial group and Q itself. Which of these subgroups are cyclic? Which of them are abelian?

Let Q be the set of eight elements, {±1, ±i, ±j, ±k}. Find List all subgroups of Q, including the trivial group and Q itself. Which of these subgroups are cyclic? Which of them are abelian?

Let Q be the set of eight elements, {±1, ±i, ±j, ±k}. Find List all subgroups of Q, including the trivial group and Q itself. Which of these subgroups are cyclic? Which of them are abelian?

Explanation / Answer

The givenset is Q = { 1, - 1-,i, - i , j ,-j , k, -k}

the subgroupws are {1} where 1 is the identity element , Q itself and the following subgroups all are abelian and cyclic

A = < i> =< - i> = { 1,-,1,i,-i} is abelian as the usual operation x satisfies commutative property ,and cyclic with 2 generators i , - i as i 1=i , i2= - 1, i3= - i ,i 4=1 . ie the element i generates all nthe elements of the subgroup A .hence it is cyclic

Similarly B = { 1,-1 j ,-j} and C = { 1,-1,k,-k}

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