Prove or disprove that Z4 is isomorphic to U5 Prove or dispr

Solution

We know that Z4 is a cylic group under addition and also U(5) under multiplication.

Since both groups are cyclic let us find a function which sends generator to generator

f(k)=2^k

i.e. f(0) = 1, f(1) = 2,...

f(a+b) = 2^a+b= 2^a*2^b

= f(a) f(b)

Hence f is a homomorphism preserving structure.

f is one to one as f(k) = f(l) implies 2^k = 2^l hence k =l

Onto also because for any 2^k we can find a k in
Z5

So proved that Z4 is isomorphic to U5