"2. We say that a group G is cyclic if there exists an element g ∈ G such that G = (g) := {gn | n ∈ Z} Given any group homomorphism φ : G H, say if each of the following is true or false, and justify. (i) If φ is surjective and G is cyclic, then H is cyclic. (ii) If φ is injective and G is cyclic, then H is cyclic. (iii) If φ is surjective and...
21. Let G be a cyclic group of order n. Show that there are exactly o(n) generators for G.
(a) Let G be a cyclic group of order n. Prove that fo every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70 -
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map. Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.
(3) (7 points) Let G be a finite abelian group of order n. Let k be relatively prime to n. Prove the map : G G given by pla) = ak is an automor- phism of G
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
(9) Let G be a group, and let x E G have finite order n. Let k and l be integers. Prove that xk = xl if and only if n divides l_ k.
Let G be a cyclic group of order 30. Find all the subgroups of G. Write the lattice of subgroups. Justif your answer and cite the theorems that allow you to determine such lattice.