Prime power order implies center is normality-large
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., group of prime power order) must also satisfy the second subgroup property (i.e., group whose center is normality-large)
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- prime power order implies not centerless: The center of a nontrivial group of prime power order is nontrivial.
- any group of prime power order is nilpotent: Any group of prime power order is nilpotent.
- Nilpotent implies center is normality-large: In a nilpotent group, the intersection of the center with any nontrivial normal subgroup is nontrivial.
Given: A group of prime power order with center , a nontrivial normal subgroup of .
To prove: is nontrivial.
Proof: Consider the action of on by conjugation. This action is well-defined because is normal in .
The fixed points under this action are the elements of that commute with every element of which is .
Suppose, under this action, the elements of that are not fixed points decompose into orbits . Then, we have:
By the fundamental theorem of group actions (fact (1)), we have:
where is the stablizer of some point in .
Since each has size greater than one, is a proper subgroup, so we have that (by fact (3), Lagrange's theorem). Since the order of is a power of and is proper, is a power, and in particular, a multiple of . Thus, we get:
Since is nontrivial, is a multiple of , hence so is . Thus, is nontrivial.