Let be a group of order . Prove that if is a normal subgroup of order in then , where represents the center of .
(Please do not use sylow's or cauchy's theorems.)
(*) If H is a subgroup of G, then the factor group is isomorphic to a subgroup of Aut H (Hungerford, "Algebra", p92).
By assumption, . Since H is a group of order 17, H is isomorphic to the cyclic group of order 17, i.e., . Thus, |Aut(H)|=16.
Now, divides both 16 and 3825 by Lagrange's theorem and (*), it follows that . Can you conclude from here?