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hw4_solutions

# hw4_solutions - Solutions to Homework 4 May 1 2010 1 We...

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Solutions to Homework 4 May 1, 2010 1. We will prove this via induction: The first term in the Upper Central Series is { e } , which is clearly a characteristic group. Suppose, for proof by mathematical induction, we have shown that { e } = Z 0 Z 1 ≤ · · · ≤ Z k are all characteristic subgroups of G . Let us now consider Z k +1 : Let ψ : G G/Z k be the obvious mapping (note that Z k is normal in G ). Notice that since Z k is characteristic, we have that σ will map the kernel of ψ into itself; in other words, ψ ( x ) = e ⇐⇒ ( ψσ )( x ) = e. (1) Let Z = Z ( G/Z k ) Then, Z is normal in G/Z k , and its inverse image via ψ is normal in G , and is what we call Z k +1 . Now suppose z Z k +1 . We will now attempt to show that ( ψσ )( z ) commutes with everything in G/Z k , which would prove ( ψσ )( z ) Z , and therefore σ ( z ) Z k +1 ; applying the same argument using σ 1 , which is also an automorphism of G , we will get σ ( Z k +1 ) = Z k +1 , thereby completing the proof of the induction step.

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