Let Z be the group of the integes. oddition. Let f:232120e tre homonorphism fiam 2oZg. defined with f(n)-4n. Find Ker f and fCZ) Ahe kemel and. with operation -> the imoge oe f).

Exercise 2.2. Let (G, *) be a group satisfying that (a* b)² = a² * 62 for every a, b = G. Show that G is abelian. Extended quesion* (Opt): What happens if the condition is replaced by (a + b)³ = a³ * 6³?

9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1).     b) find the inverse of 30 in the group U(101).