Question

Consider the group ($U_n$, $\odot_n$), where $\odot_n$ denotes $\cdot \cdot \cdot$ mod $n$, and $U_n = \{x \in \mathbb{Z}_n : gcd(x, n) = 1\}$. a) Complete the full Cayley table for each of the groups ($U_7$, $\odot_7$) and ($U_{13}$, $\odot_{13}$). (5 points each - 10 points) c) Determine the elements $a$ of ($U_7$, $\odot_7$) for which there is an $x \in U_7$ such that $a = x \odot_7 x$. (5 points) d) Determine the elements $a$ of ($U_{13}$, $\odot_{13}$) for which there is an $x \in U_{13}$ such that $a = x \odot_{13} x$. (5 points) e) For each individual $a$ as in c), determine the order of $a$. (5 points) f) For each individual $a$ as in d), determine the order of $a$. (5 points)

          Consider the group ($U_n$, $\odot_n$), where $\odot_n$ denotes $\cdot \cdot \cdot$ mod $n$, and
$U_n = \{x \in \mathbb{Z}_n : gcd(x, n) = 1\}$.
a) Complete the full Cayley table for each of the groups ($U_7$, $\odot_7$) and ($U_{13}$, $\odot_{13}$). (5 points each - 10 points)
c) Determine the elements $a$ of ($U_7$, $\odot_7$) for which there is an $x \in U_7$ such that $a = x \odot_7 x$. (5 points)
d) Determine the elements $a$ of ($U_{13}$, $\odot_{13}$) for which there is an $x \in U_{13}$ such that $a = x \odot_{13} x$. (5 points)
e) For each individual $a$ as in c), determine the order of $a$. (5 points)
f) For each individual $a$ as in d), determine the order of $a$. (5 points)

Consider the group (Un, ), where denotes ··· mod n, and
Un = {x ∈ℤn : gcd(x, n) = 1}.
a) Complete the full Cayley table for each of the groups (U7, ⊙7) and (U13, ⊙13). (5 points each - 10 points)
c) Determine the elements a of (U7, ⊙7) for which there is an x ∈ U7 such that a = x ⊙7 x. (5 points)
d) Determine the elements a of (U13, ⊙13) for which there is an x ∈ U13 such that a = x ⊙13 x. (5 points)
e) For each individual a as in c), determine the order of a. (5 points)
f) For each individual a as in d), determine the order of a. (5 points)

Added by Ashley A.

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