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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2018 • Question 3

Question

Math 6310 • Algebra Fall 2018 • Question 3

Suppose that \(p\) is the smallest prime dividing the order of a group \(G\), and that \(P\) is a Sylow \(p\)-subgroup of \(G\). If \(P\) is cyclic, show that \[ N_G(P)=C_G(P), \] i.e., that the normalizer of \(P\) in \(G\) agrees with the centralizer of \(P\) in \(G\).

Raw LaTeX

Suppose that \(p\) is the smallest prime dividing the order of a group \(G\), and that \(P\) is a Sylow
\(p\)-subgroup of \(G\). If \(P\) is cyclic, show that
\[
N_G(P)=C_G(P),
\]
i.e., that the normalizer of \(P\) in \(G\) agrees with the centralizer of \(P\) in \(G\).

Topic Tags

Finite Group Theory Normal Subgroups and Commutator Subgroups Sylow Theorems