MODERN ALGEBRA 1: HOMEWORK 5(1) Chapter 2: 7.1(2) Chapter 2: 8.1(3) LetGbe a group of orderp, wherepis a prime. Show thatGmust be isomorphictoZp.(4) Chapter 2: 8.10(5) SupposeH⊂Gis a subgroup such thataHa-1⊂Hfor alla∈G. Show thatHisnormal. (This is a trick we used in class.)(6) Give an example of a groupGand a normal subgroupH/Gsuch thatGis notisomorphic toH×(G/H). Give an example whereHandG/Hare abelian, butGis not. (It is OK to kill two birds with one stone.)Remember this example for the future.(7) Thecenter Z(G)of a groupGis the subset of elements that commute with every
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