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New page: ==Statement== Let <math>G</math> be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime <math>p</math>. Then, the subgroup <math>...
==Statement==

Let <math>G</math> be a [[nilpotent p-group]], i.e., a [[nilpotent group]] where the order of every element is a power of the prime <math>p</math>. Then, the subgroup <math>\Omega_1(Z(G))</math> is a [[normality-large subgroup]] of <math>G</math>: its intersection with every nontrivial normal subgroup is nontrivial.

Here, <math>\Omega_1</math> denotes the [[omega subgroup]]: the subgroup generated by all the elements of order <math>p</math>, and <math>Z(G)</math> denotes the [[center]] of <math>G</math>.

Note that if <math>G</math> is a finite p-group, i.e., a [[group of prime power order]], then it is nilpotent.

==Facts used==

* [[Nilpotent implies center is normality-large]]
* [[Omega-1 is large]] (and hence, is normality-large)
* [[Normality-largeness is transitive]]
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