Consider the automorphism group G = Aut(Z25, +) ∼= (U25, ·)
ID: 2985209 • Letter: C
Question
(a) What is the order of G? Is G cyclic? What are the orders of its Sylow subgroups?(b) What are the isomorphism types of the Sylow subgroups in G and how many
distinct subgroups of each type are there?
(c) Exhibit an explicit subgroup in G of each Sylow type. (List its elements in U25.)
Notes: Z25 is a commutative ring and its group of units U25 is abelian. You might want to
determine a few subgroups generated by elements of U25 to answer these questions, but won’t
need to do this for all 20 elements. (a) What is the order of G? Is G cyclic? What are the orders of its Sylow subgroups?
(b) What are the isomorphism types of the Sylow subgroups in G and how many
distinct subgroups of each type are there?
(c) Exhibit an explicit subgroup in G of each Sylow type. (List its elements in U25.)
Notes: Z25 is a commutative ring and its group of units U25 is abelian. You might want to
determine a few subgroups generated by elements of U25 to answer these questions, but won’t
need to do this for all 20 elements.
Explanation / Answer
To answer A)
The automorphisms of Z25 send 1 to a generator. A generator would be a number a such that (a,25) = 1. So the possibilities are a =1 , 2, 3, 4, 6, 7,8,9,11,12,13,14,16,17,18,19,21,22,23,24. So the order of Aut (Z25) = 20 = 2^2 * 5.
So the order of the Sylow subgroups are 2^2=4 and 5.