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???

Right?