Choose the correct statements;a)Every non-abelian group has a non-trivial abelian sub-groupb)Every non-trivial abelian group has a cyclic sub-groupc)The smallest order for a group to have a non-abelian proper sub-group is 12d)There exist a group containing elements a and b such that o(a) = 0(b) = 2 and o(ab) = 3Correct answer is option 'A,B,C,D'. Can you explain this answer?

Question

Answer

For option (a):

Let G be a nonabelian group.

Then G is not cyclic and let x∈G such that x≠e.

Also let H = (x), then H is cyclic subgroup of G.

⇒ option (a) is correct.

For option (b)

Let G be a non-trival abelian group.

If G is cyclic, then nothing to prove.

If G is non cyclic then g≠ (x) ∀∈G. Let a∈G and consider H = ��. then H is a cyclic subgroup of G.

For option (c)

Let the smallest order for a group to have a non abelian proper subgroup is n.

Clearly n≠1, 2, 3, 4, 5 because if o(G) ≤5 then G is abelian.

If o(G) = 6 then every proper subgroup of G is cyclic. So, o(G) ≠ 6.

If o(G) = 7, 11 then G is cyclic. So. o(G) ≠ 7.11.

If o(G) =8, 9, 10, then every proper subgroup of G is cyclic.

If o(G) = 12 then let G=S₃×ℤ2

⇒ G has a proper subgroup which is non abelian.

For option (d)

Let G=S₃

Let a = (1 2) and b = (2 3)

Clearly o(a) = o(b) =2

But o(ab) = 3

⇒ Statement is true

Similar Mathematics Doubts

Consider the following statements:1. A group of order 289 is abelian.2. In S3 there are four elements satisfying x2= e and six elements satisfying y3 = e.3. Every proper subgroup of S3 is cyclic.4. (Z30, has 4 subgroups of order 15 .Correct statements are

G is a group of order 51. Then which one of the following statements is false?

Choose the correct option

Which of the following statements is/are true ?

Which one of the following is false

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