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Let ''a'' belong to a group ''G'' and let |''a''| be finite. Let <math>\phi_a</math> be the automorphism of ''G'' given by <math>\phi_a (x) = axa^{-1}</math>. Show that |<math>\phi_a</math>| divides |''a''|. Exhibit an element ''a'' from a group for which 1<|<math>\phi_a</math>|<|''a''|. | Let ''a'' belong to a group ''G'' and let |''a''| be finite. Let <math>\phi_a</math> be the automorphism of ''G'' given by <math>\phi_a (x) = axa^{-1}</math>. Show that |<math>\phi_a</math>| divides |''a''|. Exhibit an element ''a'' from a group for which 1<|<math>\phi_a</math>|<|''a''|. | ||
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| + | By the properties of isomorphisms, we know that <math>\scriptstyle\phi_\alpha\phi_\beta\ =\ \phi_{\alpha\beta}</math>. Inductively then we know that <math>\scriptstyle(\phi_\alpha)^k\ =\ \phi_{\alpha^k}</math>. Let <math>\scriptstyle\mid a\mid\ =\ n</math>. Then <math>\scriptstyle a^n\ =\ e</math>, and <math>\scriptstyle\phi_{a^n}\ =\ \phi_e</math>. So, <math>\scriptstyle(\phi_a)^n\ =\ \phi_{a^n}\ =\ \phi_e</math>, and <math>\scriptstyle\mid\phi_a\mid\ \textstyle\mid\scriptstyle\ n</math>. | ||
| + | :--[[User:Narupley|Nick Rupley]] 12:15, 11 February 2009 (UTC) | ||
Revision as of 07:15, 11 February 2009
Let a belong to a group G and let |a| be finite. Let $ \phi_a $ be the automorphism of G given by $ \phi_a (x) = axa^{-1} $. Show that |$ \phi_a $| divides |a|. Exhibit an element a from a group for which 1<|$ \phi_a $|<|a|.
By the properties of isomorphisms, we know that $ \scriptstyle\phi_\alpha\phi_\beta\ =\ \phi_{\alpha\beta} $. Inductively then we know that $ \scriptstyle(\phi_\alpha)^k\ =\ \phi_{\alpha^k} $. Let $ \scriptstyle\mid a\mid\ =\ n $. Then $ \scriptstyle a^n\ =\ e $, and $ \scriptstyle\phi_{a^n}\ =\ \phi_e $. So, $ \scriptstyle(\phi_a)^n\ =\ \phi_{a^n}\ =\ \phi_e $, and $ \scriptstyle\mid\phi_a\mid\ \textstyle\mid\scriptstyle\ n $.
- --Nick Rupley 12:15, 11 February 2009 (UTC)
