# Omega-1 of center is normality-large in nilpotent p-group

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Revision as of 20:23, 7 July 2008 by Vipul (talk | contribs) (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 be a nilpotent p-group, i.e., a nilpotent group where the order of every element is a power of the prime . Then, the subgroup is a normality-large subgroup of : its intersection with every nontrivial normal subgroup is nontrivial.

Here, denotes the omega subgroup: the subgroup generated by all the elements of order , and denotes the center of .

Note that if 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