(a) Let be an element of a finite group . Define the order of and the order of . State and prove Lagrange's theorem. Deduce that the order of divides the order of .
(b) If is a group of order , and is a divisor of where , is it always true that must contain an element of order ? Justify your answer.
(c) Denote the cyclic group of order by .
(i) Prove that if and are coprime then the direct product is cyclic.
(ii) Show that if a finite group has all non-identity elements of order 2 , then is isomorphic to . [The direct product theorem may be used without proof.]
(d) Let be a finite group and a subgroup of .
(i) Let be an element of order in . If is the least positive integer such that , show that divides .
(ii) Suppose further that has index . If , show that for some such that . Is it always the case that the least positive such is a factor of ? Justify your answer.