Can a normal subgroup of a transitive group also be transitive?

  • Context: Undergrad 
  • Thread starter Thread starter Euge
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    2015
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SUMMARY

The discussion centers on the problem of whether a nontrivial normal subgroup \( A \) of a transitive group \( G \) within the symmetric group \( S_p \) is also transitive. The conclusion, as proven by user johng, is that \( A \) must indeed be a transitive subgroup of \( S_p \). This result is significant in group theory, particularly in understanding the properties of normal subgroups within transitive groups.

PREREQUISITES
  • Understanding of group theory concepts, particularly normal subgroups.
  • Familiarity with transitive groups and their properties.
  • Knowledge of symmetric groups, specifically \( S_p \).
  • Basic proof techniques in abstract algebra.
NEXT STEPS
  • Study the properties of normal subgroups in group theory.
  • Explore the implications of transitivity in symmetric groups.
  • Learn about the structure of \( S_p \) and its subgroups.
  • Investigate examples of transitive groups and their normal subgroups.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra and group theory, as well as students seeking to deepen their understanding of transitive groups and normal subgroups.

Euge
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Here is this week's POTW:

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Let $p$ be a prime, $G$ a transitive subgroup of the symmetric group $S_p$, and $A$ a nontrivial normal subgroup of $G$. Prove that $A$ is a transitive subgroup of $S_p$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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This week's problem was correctly solved by johng. Here is his solution:
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