Normal group contained in the center.

In summary, the conversation discusses the topic of verifying a write-up on group theory and seeking tips for more efficient methods. The conversation also mentions the Orbit-Stabilizer Theorem and its application in proving that a normal subgroup of order p in a group of order p^n is in the center of the group. The solution involves constructing a group action and using the Orbit-Stabilizer Theorem to show that all elements of the subgroup are in the center of the group.
  • #1
Barre
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I am doing exercises from Hungerford's text 'Algebra', and would appreciate if someone took the time to verify my write-up for me, and possibly provide me with tips how this could be done more efficiently (using less mathematical machinery)

Homework Statement



If a normal subgroup N of order p ( p prime) is contained in a group G of order [itex]p^n[/itex], then N is in the center of G.

Homework Equations


Orbit-Stabilizer Theorem: http://www.proofwiki.org/wiki/Orbit-Stabilizer_Theorem

The Attempt at a Solution


I construct the group action [itex]f: G \times H \rightarrow H[/itex] by this following rule: [itex]f((g,h)) = ghg^{-1}[/itex]. This is well-defined by normality of H of course.
By Orbit-Stabilizer Theorem I know that the size of orbit of any [itex]h \in H[/itex] must have cardinality dividing [itex]|G| = p^n[/itex]. So cardinality of orbits is 1 or p (since anything bigger
would imply more elements in an orbit than there are in H) and H is the disjoint union of orbits of it's elements, so all orbits cardinalities add up to p. But we see that the orbit of the identity in H must be of size 1 (itself), since for any [itex]g \in G[/itex] , [itex]geg^{-1} = e[/itex]. This means we have p-1 other elements in orbits, but orbit cardinalities have to divide p, so they are all of size 1.
This means that for any [itex] h \in H[/itex] and all [itex] g \in G[/itex] we have [itex] ghg^{-1} = h[/itex], so all elements of H are in the center of G.
 
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  • #2
That's very nicely written. And the Orbit-Stabilizer Theorem is about the only piece of machinery you used. I can't think how you would do it more simply.
 

1. What is a normal group contained in the center?

A normal group contained in the center is a subgroup of a larger group that is contained within the center of the larger group. This means that every element of the subgroup commutes with every element of the larger group.

2. How is a normal group contained in the center different from a regular normal group?

A normal group contained in the center is a special kind of normal group where the subgroup is contained within the center of the larger group. In a regular normal group, the subgroup does not necessarily have to be contained within the center.

3. What are the properties of a normal group contained in the center?

A normal group contained in the center has the properties of being both a normal subgroup and a central subgroup. This means that it is invariant under conjugation and that all of its elements commute with all elements of the larger group.

4. How can a normal group contained in the center be used in group theory?

A normal group contained in the center is useful in group theory for studying the structure of a larger group. It can help identify important properties and relationships within the group and can be used to prove theorems and solve problems.

5. Can a normal group contained in the center be a trivial subgroup?

Yes, a normal group contained in the center can be a trivial subgroup, meaning it only contains the identity element. However, it can also be a non-trivial subgroup and still satisfy the properties of being normal and contained in the center.

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