Question

Could you please solve the following question using the hint (group actions)? Thank you so much! Find all automorphisms of order 3 of 291 (Hint: How can 23 act non-trivially on Z91)

Answer 1

Note that 91 = 7·13, and by our study of groups of order pq, we know that Z_{​​​​​91}$\congruent$$\cong$ Z₇×Z₁₃. Thus,
we have that Aut(Z_{​​​​​91}) $\cong$ Aut(Z₇× Z₁₃). Let Z₇ × Z₁₃be generated by the element {x} × {y}.

An automorphism of a cyclic group is completely determined by where it sends x and y to. Since
7 and 13 are prime, we can send x to six different generators and y to twelve different generators.
Thus Aut(Z₇ × Z₁₃) $\cong$ Z₆ × Z₁₂.

Now that we have this isomorphism, we can find the automorphisms of order 3 by identifying

the elements of order 3 in Z₆ × Z₁₂. Then Z₆ has two elements of order three corresponding
to the automorphisms [
T 4.2
] and [$x \mapsto x^4$], and Z₁₂also has two elements of order three
corresponding to the automorphisms [
уну
] and [$y\mapsto y^9$].

Thus, the number of automorphisms of order 3 is 8: all combinations of the identity
or automorphisms of order three gives us 9 automorphisms, and we subtract the identity
automorphism which has order 1, leaving us with 9 − 1 = 8 automorphisms of order 3.
On Z₇× Z₁₃ these are the automorphisms:

(x, y) $\mapsto$ (x, y³)
(x, y) $\mapsto$ (x, y⁹)
(x, y)$\mapsto$ (x², y)
(x, y)$\mapsto$ (x², y³)
(x, y) $\mapsto$ (x², y⁹⁾
(x, y) $\mapsto$ (x⁴, y)
(x, y)$\mapsto$ (x⁴ ,y²⁾
(x, y)$\mapsto$ (x⁴, y⁸)
To find the correspondence between these automorphisms of Z_7* Z₁₃and the automorphisms
of Z₉₁, we pick a generator a such that Z₉₁= <a>, and solve the congruences

m ≡ i mod 7
m ≡ j mod 13 ,
for i ∈ {1, 2, 4}, j ∈ {1, 3, 9}, and i, j both not 1. Then, each m gives us an automorphism of
Z₉₁ [a$\mapsto$ a^m] which has order 3. The solutions are:

m ≡ 1 mod 7
m ≡ 3 mod 13
  $\mapsto$m = 29

m ≡ 1 mod 7
m ≡ 9 mod 13
$\mapsto$ m = 22

m ≡ 2 mod 7
m ≡ 1 mod 13
$\mapsto$ m = 79

m ≡ 2 mod 7
m ≡ 3 mod 13

$\mapsto$ m = 16

m ≡ 2 mod 7
m ≡ 9 mod 13
$\mapsto$ m = 9

m ≡ 4 mod 7
m ≡ 1 mod 13

$\mapsto$m = 53

m ≡ 4 mod 7
m ≡ 3 mod 13
$\mapsto$ m = 81

m ≡ 4 mod 7
m ≡ 9 mod 13
$\mapsto$ m = 74
So, the automorphisms of order 3 of Z₉₁ are

a $\mapsto$ a⁹

a $\mapsto$ a¹⁶

a$\mapsto$ a²²
a $\mapsto$ a²⁹
a $\mapsto$ a⁵³
a $\mapsto$ a⁷⁴
a  $\mapsto$a⁷⁹
a $\mapsto$ a^81.