Given a set Z$13=\{0,1,2,3,4,5,6,7,8,9,10,11,12\},$ we say that Z$13$ is an additive group mod $13.$

$($a$)$ What is the identity of this group? and what is the order of this group?

$($b$)$ We define a function fe : Z$13->$ Z$13,$ where fe$($g$)=$ e$$g$,$ e is an integer and g is an element of this group Z$13.$ If we choose e $=3,$ then is this function fe a permutation of this group Z$13?$ If it is a permutation when e $=3,$ then please explain why and compute the output of each input using this function fe$.$

$($c$)$ Based on this additive group Z$13,$ please list all the elements in multiplicative group $.$ What is the identity of group and what is the order of $?$

$($d$)$ Please list all the elements in group Z$20$ and all the elements in group