\b[No.] The alternating group $A_4$ has order twelve, and six is a proper divisor of twelve, but $A_4$ has no element of order six. Its nonidentity elements are eight 3-cycles of order three and three double transpositions of order two.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 5D / b / Solution

 Solution (source code)