Question

Problem 2 Part a Prove that for $U(13)$ prove that the function $\phi_5: U(13) \to U(13)$ defined by the formula $\phi_5(x) = x^5$ is an automorphism of $U(13)$. Part b Determine the integers $n$ less than or equal to 12 such that $\phi_n(x) = x^n$ is an automorphism of $U(13)$. You do not need to give a complete proof of your answer, but you should give at least enough work to determine a pattern. Part c Based on your work in Part a and Part b, state a general claim about when, for an integer $n$ and a prime $p$, the function $\psi_n: U(p) \to U(p)$ defined by $\psi_n(x) = x^n$ is an automorphism of $U(p)$.

          Problem 2
Part a
Prove that for $U(13)$ prove that the function
$\phi_5: U(13) \to U(13)$ defined by the
formula $\phi_5(x) = x^5$ is an automorphism of
$U(13)$.
Part b
Determine the integers $n$ less than or equal to
12 such that $\phi_n(x) = x^n$ is an
automorphism of $U(13)$. You do not need to
give a complete proof of your answer, but you
should give at least enough work to
determine a pattern.
Part c
Based on your work in Part a and Part b, state
a general claim about when, for an integer $n$
and a prime $p$, the function
$\psi_n: U(p) \to U(p)$ defined by
$\psi_n(x) = x^n$ is an automorphism of $U(p)$.

Problem 2
Part a
Prove that for U(13) prove that the function
ϕ5: U(13) → U(13) defined by the
formula ϕ5(x) = x^5 is an automorphism of
U(13).
Part b
Determine the integers n less than or equal to
12 such that (x) = x^n is an
automorphism of U(13). You do not need to
give a complete proof of your answer, but you
should give at least enough work to
determine a pattern.
Part c
Based on your work in Part a and Part b, state
a general claim about when, for an integer n
and a prime p, the function
: U(p) → U(p) defined by
(x) = x^n is an automorphism of U(p).

Added by Jennifer G.

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