Difference between revisions of "Omega1 of center is normalitylarge in nilpotent pgroup"
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(New page: ==Statement== Let <math>G</math> be a nilpotent pgroup, i.e., a nilpotent group where the order of every element is a power of the prime <math>p</math>. Then, the subgroup <math>...) 

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Revision as of 20:23, 7 July 2008
Statement
Let be a nilpotent pgroup, i.e., a nilpotent group where the order of every element is a power of the prime . Then, the subgroup is a normalitylarge 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 pgroup, i.e., a group of prime power order, then it is nilpotent.
Facts used
 Nilpotent implies center is normalitylarge
 Omega1 is large (and hence, is normalitylarge)
 Normalitylargeness is transitive