# Group theory cycles question error

In a solution to a question, there is a line that reads in the group $$S_{10}$$: $$(1,8,2,5)^2 = (1,2,8,5)$$.

It is surely not meant to read: $$(1,8,2,5)^2 = (1,2)(5,8)$$ or am I mistaken about how cycles work?

• No, you're right. It must be a typo. – Bernard May 12 at 17:17

Indeed, $$(1825)^2 = (12)(58)$$, and the solution has a typo. You can check that it is a typo quickly by seeing that as $$4$$-cycles have order $$4$$, squaring any $$4$$-cycle must give an element of order $$2$$, and so one can never obtain a $$4$$-cycle by squaring a $$4$$-cycle.