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
Links
- Wikipedia, Lagrange's theorem (group theory)
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