I haven't put much thought into these problems, so it's possible that some of them may be too hard or too easy. Every quotient group of a finite group is again of finite order. Every quotient group of an Abelian group is Abelian. There is a homomorphism from a group of 6 elements into a group of 12 elements. A homomorphism is one to one if and only if its kernel consists of just the identity.
Every homomorphism is one to one.
A homomorphism may have an empty kernel. Every factor group of a cyclic group is cyclic. A divisor of 0 in a commutative ring with unity can have no multiplicative inverse. Every field is also a ring. The nonzero elements of a field for a group under multiplication.