There are only 2.

define for

and for

check that these are homomorphisms

also, they will be the only ones. since any homomorphism has to be of the form for some integer n. this only works with n = +/- 1, otherwise the map would not be onto.

from directly above, we can choose n to be any integer here, so there are an infinite amount.How many homomorphisms are there of Z into Z?

there are 2.How many homomorphisms are there of Z into Z_2?

for all

or define a mapping based on , in which case

Z_2 x Z_12 = Z_2 x Z_3 x Z_4Are the following groups isomorphic?

Z_2 X Z_12 and Z_4 X Z_6

I think these are because the lcm(2,12)=lcm(4,6). Would this be a right assumption?

and Z_4 x Z_6 = Z_4 x Z_2 x Z_3

these are the same thing, just rearranged. thus they are isomorphic

the orders are different. thus they are not isomorphicZ_8 X Z_10 X Z_24 and Z_4 X Z_10 X Z_40

I think these are because the lcm(8,10,24)=lcm(4,12,40)