The document discusses conjugate cyclic permutations in group theory. It defines a cyclic permutation τ as a bijection from a set of numbers n to itself, where it maps n to n and a number k to the number following it. It then defines the conjugate of a cyclic permutation τ by another cyclic permutation τ-1 as mapping the elements i1, i2, ..., ir of τ to their images under τ-1 in the same order. So the conjugate of τ by τ-1 is obtained by applying τ-1 to each element of τ.

1. Conjugate
cyclic permutation
June 15, 2016 (Wednesday) 13:46 (Japan time)
Nagaoka University of Technology
M1 Tamura Takumi

2. 巡回置換の共役変換
𝜏 ∈ 𝑆 𝑛 = 𝜎: 𝑛 → 𝑛 | 𝐵𝑖𝑗𝑒𝑐𝑡𝑖𝑜𝑛 , 𝑛 = 1,2, … , 𝑛 − 1, 𝑛
𝜏 [𝑛] = 𝑛 , 𝑘 ∈ 𝑛
𝜏 𝑖1, 𝑖2, … , 𝑖 𝑟 𝜏−1 𝜏 𝑘 = 𝜏 𝑖1, 𝑖2, … , 𝑖 𝑟 𝑘 =
𝜏 𝑖𝑗+1 mod 𝑟 𝑖𝑓 𝑘 = 𝑖𝑗 ∈ {𝑖1, 𝑖2, … . , 𝑖 𝑟}
𝑘 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
∴ 𝜏 𝑖1, 𝑖2, … , 𝑖 𝑟 𝜏−1
= 𝜏 𝑖1 , 𝜏 𝑖2 , … , 𝜏(𝑖 𝑟)