Mark each of the following true or false. 

___a. Every homology group of a contractible space is the trivial group of one element. 

___b. A continuous map from a simplicial complex X into a simplicial complex Y induces a homomorphism of H_n(X) into H_n(Y). 

___ c. All homology groups are abelian. 

___ d. All homology groups are free abelian. 

___ e. All 0-dimensional homology groups are free abelian. 

___ f. If a space X has n-simplexes but none of dimension greater than n and H_n(X) ≠ 0, then H_n(X) is free abelian. 

___ g. The boundary of an n-chain is an (n - 1)-chain. 

___ h. The boundary of an n-chain is an (n - 1)-cycle. 

___ i. Then-boundaries form a subgroup of the n-cycles. 

___ j. The n-dimensional homology group of a simplicial complex is always a subgroup of the group of n-chains.