Jelly, 8 bytes
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Outputs 0 for False and a non-negative integer for True
Jelly finally ties J!
How it works
JịƬ"ZẈiL - Main link. Takes the Cayley matrix n×n M on the left
Z - Transpose M
J - Yield [1, 2, ..., n]
Ƭ - Until reaching a fixed point, do the following and replace i with the result