Claim: If G has no subgroups H /= (e), G, then G must be cyclic of prime order.
Proof:
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 g 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.
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.
