Difference between revisions of "Omega1 of center is normalitylarge in nilpotent pgroup"
From Groupprops
(No difference)

Revision as of 17:42, 3 September 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