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hw5.pdf - MODERN ALGEBRA 1 HOMEWORK 5(1 Chapter 2 7.1(2...

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MODERN ALGEBRA 1: HOMEWORK 5 (1) Chapter 2: 7.1 (2) Chapter 2: 8.1 (3) Let G be a group of order p , where p is a prime. Show that G must be isomorphic to Z p . (4) Chapter 2: 8.10 (5) Suppose H G is a subgroup such that aHa - 1 H for all a G . Show that H is normal. (This is a trick we used in class.) (6) Give an example of a group G and a normal subgroup H / G such that G is not isomorphic to H × ( G / H ) . Give an example where H and G / H are abelian, but G is not. (It is OK to kill two birds with one stone.) Remember this example for the future. (7) The center Z ( G ) of a group G is the subset of elements that commute with every
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