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

  • MHB
  • Thread starter Euge
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    2015
In summary, a normal subgroup of a transitive group can also be transitive. A transitive group is one that acts transitively on a set, meaning it can move any element to any other element. The normality of a subgroup does not necessarily affect its transitivity, as it may still act transitively on a set. However, not all normal subgroups of a transitive group are transitive. A group cannot be both transitive and non-transitive at the same time.
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Euge
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MHB
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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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  • #2
This week's problem was correctly solved by johng. Here is his solution:
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FAQ: Can a normal subgroup of a transitive group also be transitive?

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

Yes, it is possible for a normal subgroup of a transitive group to also be transitive. This is because the definition of a transitive group only requires that the group acts transitively on a set, not that all subgroups must also be transitive.

What does it mean for a group to be transitive?

A transitive group is a group that acts transitively on a set, meaning that for any two elements in the set, there exists an element in the group that can map one element to the other. In other words, the group's action on the set is able to move any element to any other element.

How does a subgroup being normal affect its transitivity?

If a subgroup is normal, it means that it is invariant under conjugation by any element in the group. This does not necessarily affect its transitivity, as the subgroup can still act transitively on a set, but it may have additional properties that differ from the larger group.

Are all normal subgroups of a transitive group also transitive?

No, not all normal subgroups of a transitive group are necessarily transitive. While it is possible for a normal subgroup to also be transitive, this is not always the case. It depends on the specific group and subgroup in question.

Can a group be both transitive and non-transitive?

No, a group cannot be both transitive and non-transitive at the same time. A group is either transitive, meaning it acts transitively on a set, or it is non-transitive, meaning it does not act transitively on any set. It cannot possess both properties simultaneously.

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