find a homomorphism between additive groups Z3 and Z6 Solution if ? is a homomor
ID: 3080543 • Letter: F
Question
find a homomorphism between additive groups Z3 and Z6Explanation / Answer
if ? is a homomorphism, ?(Z3) has to be a subgroup of Z6. furthermore, ker(?) has to be a subgroup of Z3. now Z3 has just two subgroups: {0} and {0,1,2}. so either ker(?) = {0}, or ker(?) = Z3. case 1: ker(?) = {0}. in this case, ? is 1-1, so ?(Z3) has to be a 3-element subgroup of Z6. the only 3-element subgroup of Z6 is = {0,2,4}. so ?:k-->2k is one homomorphism. (it should be clear this IS a homomorphism, in fact ?(Z3) is isomorphic to Z3). what about ?:k-->4k (where the 2nd number is mod 6)? convince yourself this is also a homomorphism. case 2: ker(?) = Z3. in this case ? maps everything to 0, it is the trivial map (which is always a homomorphism). that's all the possibilities. @jtabbsvt: you forgot one, and you got one wrong. suppose ?:0-->0, 1-->4, 2-->2 then ?(2+2) = ?(1) = 4 = 2 + 2 = ?(2) + ?(2) = , they are isomorphic as groups, so there is an isomorphism between them, so since there is an isomorphism between Z3 and , composing the two isomorphims gives an isomorphism between Z3 and , which sends the generator 1 to the generator 4.Related Questions
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