I have a test Wednesday and I'm working on the practice test, just wanna know if these are right, I've done them all

1)

a)list all of the elements of z_{30}that have order 10

answer: {3, 9, 21, 27}

b)let g be a group, x ϵ G, with order of x = 40. List all elements of <x> with order 10.

answer: x^{4}x^{12}x^{28}x^{36}

2)consider z_{25}. show that the map ϕ: z_{25 }-> z_{25}give by ϕ(x)=2x is an automorphism of z_{25 }

answer:

operation perserving

ϕ(x+y)

2(x+y) mod 25

2x mod 25 + 2y mod 25

ϕ(x)+ϕ(y)

As for the bijective step, I know I can show the mapping for every element but is this right also?

1:1

ϕ(x)=ϕ(y)

2x=2y

x=y

onto

let y = 2x

ϕ(x)=2x=y

Thanks!