Mark each of the following true or false. 

___ a. A,, is a normal subgroup of S_n .

___ b. For any two groups G and G', there exists a homomorphism of G into G'. 

___ c. Every homomorphism is a one-to-one map. 

___ d. A homomorphism is one to one if and only if the kernel consists of the identity element alone. 

___ e. The image of a group of 6 elements under some homomorphism may have 4 elements.

___ f. The image of a group of 6 elements under a homomorphism may have 12 elements. 

___ g. There is a homomorphism of some group of 6 elements into some group of 12 elements. 

___ h. There is a homomorphism of some groups of 6 elements into some group of 10 elements. 

___ i. A homomorphism may have an empty kernel. 

___ j. It is not possible to have a nontrivial homomorphism of some finite group into some infinite group.