proof of the converse of Lagrange’s theorem for finite cyclic groups
The following is a proof that, if G is a finite cyclic group and n is a nonnegative integer that is a divisor
of |G|, then G has a subgroup
of order n.
Proof.
Let g be a generator of G. Then |g|=|⟨g⟩|=|G|. Let z∈ℤ such that nz=|G|=|g|. Consider ⟨gz⟩. Since g∈G, then gz∈G. Thus, ⟨gz⟩≤G. Since |⟨gz⟩|=|gz|=|g|gcd(z,|g|)=nzgcd(z,nz)=nzz=n, it follows that ⟨gz⟩ is a subgroup of G of order n.
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Title | proof of the converse of Lagrange’s theorem for finite cyclic groups |
---|---|
Canonical name | ProofOfTheConverseOfLagrangesTheoremForFiniteCyclicGroups |
Date of creation | 2013-03-22 13:30:27 |
Last modified on | 2013-03-22 13:30:27 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 10 |
Author | Wkbj79 (1863) |
Entry type | Proof |
Classification | msc 20D99 |
Related topic | CyclicRing3 |
Related topic | ProofThatGInGImpliesThatLangleGRangleLeG |