In S_4, find a cyclic subgroup of order 4 and a non-cyclic one as well.
Let a and b belong to S_n. Prove that bab^-1 and a are both even or both odd.
$\displaystyle V = \{(1), (12)(34),(13)(24),(14)(23)\}$.
And $\displaystyle \left< (1234) \right>$ are such examples.
Hint: If $\displaystyle a=(a_1,...,a_k)$ then $\displaystyle bab^{-1} = (b(a_1),...,b(a_k))$.Let a and b belong to S_n. Prove that bab^-1 and a are both even or both odd.