Editing Exotic sphere (section)

===Order of Θ<sub>n</sub>===
The order of the group <math>\Theta_n</math> is given in this table {{OEIS|id=A001676}} from {{harv|Kervaire|Milnor|1963}} (except that the entry for <math>n = 19</math> is wrong by a factor of 2 in their paper; see the correction in volume III p.&nbsp;97 of Milnor's collected works).

:{| class="wikitable" style="text-align:center"
|-
! Dim  n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|-
! order <math>\Theta_n</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
|-
!<math>bP_{n+1}</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1
|-
!<math>\Theta_n/bP_{n+1}</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24
|-
!<math>\pi_n^S/J</math> 
| 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2×2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24
|-
!index
| – || 2 || – || – || – || 2 || – || – || – || – || – || – || – || 2 || – || – || – || – || – || –
|}

Note that for dim <math>n = 4k - 1</math>, then <math>\theta_n</math> are <math>28 = 2^2(2^3-1)</math>, <math>992 = 2^5(2^5 - 1)</math>, <math>16256 = 2^7(2^7 - 1) </math>, and <math>523264 = 2^{10}(2^9 - 1)  </math>. Further entries in this table can be computed from the information above together with the table of [[stable homotopy groups of spheres]].

By computations of stable homotopy groups of spheres, {{harvtxt|Wang|Xu|2017}} proves that the sphere {{math|''S''<sup>61</sup>}} has a unique smooth structure, and that it is the last odd-dimensional sphere with this property – the only ones are {{math|''S''<sup>1</sup>}}, {{math|''S''<sup>3</sup>}}, {{math|''S''<sup>5</sup>}}, and {{math|''S''<sup>61</sup>}}.