Question

prove if alpha is a k-cycle, with k>2, then alpha composed with alpha is a cylcle iff k is odd

Answer 1

i do not know a fancy proof of this but it is pretty clear yes? if k is even, say
\[\alpha = (a_1 a_2 ... a_{2j})\] then
\[\alpha^2 = (a_1 a_3 ,,,a_{2j-1}) (a_2 a_4... a_{2j})\] whereas if k is odd say
\[\alpha=(a_1 a_2 ... a_{2j+1})\] then
\[\alpha^2=(a_1 a_3 ...a_{2j+1} a_2 a_4... a_{2j})\]
try if for small values of k and it will be clear.
\[(a_1a_2a_3a_4a_5a_6)^2 = (a_1a_3a_5)(a_2a_4a_6)\] whereas
\[(a_1a_2a_3a_4a_5)^2=(a_1a_3a_5a_2a_4)\]