Proving the statements:

(a) Every cyclic subgroup of H is finite:

To prove this, let's consider an arbitrary cyclic subgroup of H.

Definition of a cyclic subgroup: A subgroup generated by a single element.

Definition of H: H is the quotient group Q/Z, which is the set of all cosets of Z in Q.

Properties of cosets: Every coset of Z in Q is of the form a + Z, where a is an element of Q.

Properties of the quotient group: The set of cosets, denoted as Q/Z, forms a group under the operation of coset multiplication.

Finite order of the cosets: Each coset a + Z has infinite elements, but the quotient group Q/Z is formed by the collection of all cosets, and there are countably infinite cosets.

Finite order of the cyclic subgroup: Since each coset has infinite elements, the cyclic subgroup generated by any coset will also have infinite elements. Therefore, every cyclic subgroup of H is infinite.

(b) Every finite cyclic group is isomorphic to a subgroup of H:

To prove this, let's consider an arbitrary finite cyclic group G.

Definition of a finite cyclic group: A group generated by a single element with a finite number of elements.

Isomorphism: An isomorphism is a bijective function that preserves the group structure.

Mapping to H: We can define a mapping from G to H by sending each element of G to its corresponding coset in H. Since each element of G is generated by a single element, it can be represented as g^n for some integer n.

Mapping function: The mapping function f: G -> H is defined as f(g) = g + Z, where g is an element of G and Z is the subgroup of integers in H.

Bijective property: The mapping function f is bijective because each element of G is mapped to a unique coset in H, and each coset in H is reached by some element of G.

Preservation of group structure: The mapping function f preserves the group structure because the operation of coset multiplication in H corresponds to the operation of group multiplication in G.

Isomorphism of G and H: Therefore, the finite cyclic group G is isomorphic to a subgroup of H through the mapping function f.