Hello everyone. I received a take-home exam for my algebra class. I was doing the problems and I think I found a typo in the exam. The question is as follows:
Let be a homomorphism and show 1, 2, or 3.
Now I though that is a subgroup of and hence, by Lagrange theorem, must divide 3003.
Now can have at most 6 elements (i.e. the order of ). But the only numbers less than or equal to 6 that divide 3003 are 1 and 3.
Also, if implies maps two elements to the identity of . This implies that the kernel has order 2. But the kernel of a homomorphism is a normal subgroup, and has no normal subgroup of order 2.
So is the only possibility???