Question

Any group of order 4 is isomorphic to either C4 = {(1), (1234), (13)(24), (1432)}, the cyclic group of order 4, or K4 = {(1), (12)(34),(13)(24),(14)(23)}, the Klein-4 group (you don't need to prove this). Does there exist an onto homomorphism from D, onto C4? Does there exist an onto homo morphism from De onto K ? Justify your answers by either explicitly giving such a homomorphism, or proving that such a homomorphism cannot exist.

Answer 1

3 10 17 24 Let D= {e, p, p2, p3, S, SP, SP? , SP3} We know that DA has 4 proper normal subgroups. and three of them have indea 2. and they are se, p, r2,837 {e, sr, r2, Sr2 şers, 5827 - in this Case the homomorphic image is 2/27 and there is one inden 4 normal subgroup seiphy.. Now, I Dal = 23 and let 2 (DA) = Center of Da then 2 divides order of 2(04). - norm we YOU momoy ( norm Now 2(04) #8, otherwise D4 is commutative. 2 1D42 & 4, otherwise, D4/2(04) oro isomorphic & 2/22, which is cyclic and Da will commutative. So, Z (D4) = {e, p2} ( since 2009 is normal sub grou of Dal And we know that Kernal of group homomorphirm is a normal subgroup. There so Therefore, there does not exists any onto homomorphism from Da to (A since, CA id Cyclic soon otherwise C4 E 8 D4/2(14), and Da io commutative] ores and there Exists hom onto homo morphiren from Du tok h. Do ka verify this is homorncorption) attal