Answer to Question #17371 in Abstract Algebra for steve peters

Question

Let f:G1-->G2 be a homomorphism of groups. Let H be a cyclic subgroup of G1. Prove the f(H) is a cyclic subgroup of G2.

Answer 1

If H cyclic then exist g from H generator element suchthat every element h from H h=g^n for some natural n
f(H)={f(h) | h belongs to H}={f(g^n)|n - natural}= | f -homomorphism so |= { (f(g))^n | n - natural}
f(H) is a subgroup of G2 and it is generated by f(g) soit is cyclic