Let . For which integers does the map defined by extend to a well defined homomorphism ? Can be a surjective homomorphism?
We begin with a couple of lemmas.
Lemma 1: Let be a cyclic group, a group, and a group homomorphism. Then is uniquely determined by . Proof: Suppose we have two homomorphisms , such that . Since is cyclic, an arbitrary can be written as for some . Then . So .
Lemma 2: Let be a group, , , and a set mapping. Then extends to a homomorphism if and only if . Proof: We have , so that . Let be integers such that . Then mod , so that for some . Thus , so that is well defined. Now suppose with . Then . Thus is a homomorphism.
Lemma 3: A map given by extends to a group homomorphism if and only if divides . Proof: If extends, then by lemma 2, we have mod , so that divides . let and write and . Then divides , and since and are relatively prime, divides by Euclid’s lemma. Conversely, if divides , then divides , so that extends to a homomorphism by Lemma 2.
Here, and . So extends if and only if . There are 12 such .
Suppose some such is surjective; then we have mod 36, a contradiction.
Note that since , . Hence cannot be surjective.