series_homework

series_homework - subgroups of G . 2. Assume G is a Fnite...

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Homework on Derived Series, and Upper and Lower Central Series April 12, 2010 1. We say that a subgroup H of a group G is characteristic if every auto- morphism σ : G G maps H H . Note that if G has only a single subgroup H of some order n , then H is characteristic. Show that the elements of the upper central series are all characteristic
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Unformatted text preview: subgroups of G . 2. Assume G is a Fnite group. Using the characterization of nipotent from class, prove that G is nilpotent if and only if G has a subgroup of each order dividing | G | . 3. ind the upper and lower central series for A 4 and S 4 . 1...
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