A361871 The smallest order of a non-abelian group with an element of order n.

6, 6, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 1

Author

Yue Yu, Apr 01 2023

Keywords

• nonn
• easy

Links

• Wikipedia, Lagrange's theorem (group theory)

Crossrefs

Essentially the same as A163300, A103517, A051755.

Formula

a(n) = 2*n for n >= 3.

Proof: By Lagrange's theorem in group theory we have that n divides a(n) for all n. A group of order n and with an element of order n is the cyclic group of order n, hence being abelian. On the other hand, the dihedral group D_{2n} is non-abelian for n >= 3 and contains an element of order n. - Jianing Song, Aug 11 2023