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.

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I think that a little simpler explanation is in (http://wolfweb.unr.edu/homepage/naik/classes/731/60.9.Soln.pdf). Just checking clarifies the desired homomorphisms, not need to care about order of or Lagrange’s Theorem.

That’s much better. Thanks!