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?
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)