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Show that a finite group G of cardinal pαm with (m, p

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Algebra I. Homework 2. Due on September 24, 2020.

1. LetPn(k)be the number of permutations of{1,2, . . . n}having exactlykfixed points. Show that Pn

k=0kPn(k) =n!.

2. Give an example of a finite group G and a divisor m of #Gsuch that G has no subgroup of orderm (hint: consider A4, #G= 12 and m= 6)

3. Show that a finite group G of cardinal pαm with (m, p) = 1 has a subgroup of order pi for any i≤α. (hint: argue by induction)

4. LetGbe a group of order255. Show thatGis not simple (hint: use ap-Sylow of Gfor some p)

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