Mark each of the following true or false. 

___ a. Every factor group of a cyclic group is cyclic. 

___ b. A factor group of a noncyclic group is again noncyclic. 

___ c. R/Z under addition has no element of order 2. 

___ d. R/Z under addition has elements of order n for all n ∈ Z⁺. 

___ e. R/Z under addition has an infinite number of elements of order 4. 

___ f. If the commutator subgroup C of a group G is { e}, then G is abelian. 

___ g. If G / H is abelian, then the commutator subgroup of C of G contains H. 

___ h. The commutator subgroup of a simple group G must be G itself. 

___ i. The commutator subgroup of a nonabelian simple group G must be G itself. 

___ j. All nontrivial finite simple groups have prime order.