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 November 11th, 2014, 05:32 PM #1 Newbie   Joined: Feb 2014 Posts: 6 Thanks: 1 Help with a problem of isomorphisms. Let $H\leq G$ and are $N(H)$, $C(H)$ and $I(H)$ the normalizer of $H$, the centralizer of $H$ and the group of all inner automorphisms of $H$, respectively. Show that $N(H)/C(H)\cong I(H)$. Last edited by robinhg; November 11th, 2014 at 05:34 PM. January 26th, 2015, 07:59 AM #2 Senior Member   Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra Let $\phi_g\in\mathrm I(G)$ be the inner automorphism induced by $g\in G$ i.e. $\phi_g(x)=gxg^{-1}$ for all $x\in G$. Now define $f:\mathrm N(H)\to\mathrm I(H)$ by $f(n)=\phi_n$ for $n\in\mathrm N(H)$ (check that this definition makes sense) and show that $f$ is a surjective homomorphism with kernel $\mathrm C(H)$. Tags automorphisms, group theory, isomorphisms, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Artus Abstract Algebra 4 January 2nd, 2013 10:36 AM Kappie Abstract Algebra 3 March 1st, 2012 01:11 PM Hyperreal_Logic Applied Math 2 January 27th, 2010 07:25 AM riemannsph12 Abstract Algebra 0 November 13th, 2008 02:11 PM Jamers328 Abstract Algebra 5 November 6th, 2008 08:21 PM

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