How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Groups with no subgroups

Claim: If G has no subgroups H /= (e), G, then G must be cyclic of prime order.


One Line Proof: If the order of G is composite, then it has Sylow Subgroups.

More than one Line Proof: If the order of G is composite then there exists d that divides |G| and 1 < d < |G|. Pick any element from G. Note that ( g^d \neq e ) otherwise we will find a nontrivial subgroup. Then consider the non-trivial subgroup generated by ( g^d ). As  ( g^{d \times \frac{|G|}{d} } = e ) hence we find a non trivial subgroup.

Therefore the order of G cannot be composite.

Rest is left as an exercise.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.