# Difference between revisions of "Omega-1 of center is normality-large in nilpotent p-group"

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

+ | |||

+ | ==Related facts== | ||

+ | |||

+ | ===Corollaries=== | ||

+ | |||

+ | * [[Minimal normal implies contained in Omega-1 of center for nilpotent p-group]] | ||

+ | * [[Socle equals Omega-1 of center for nilpotent p-group]] | ||

+ | * [[Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group]] | ||

+ | |||

+ | ===Other related facts=== | ||

+ | |||

+ | * [[Minimal normal implies central in nilpotent]] | ||

+ | * [[Minimal characteristic implies central in nilpotent]] | ||

==Facts used== | ==Facts used== | ||

− | + | # [[Nilpotent implies center is normality-large]] | |

− | + | # [[Omega-1 is large]] (and hence, is normality-large) | |

− | + | # [[Normality-largeness is transitive]] |

## Latest revision as of 13:46, 23 September 2008

## 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.

## Related facts

### Corollaries

- Minimal normal implies contained in Omega-1 of center for nilpotent p-group
- Socle equals Omega-1 of center for nilpotent p-group
- Minimal characteristic implies contained in Omega-1 of center for nilpotent p-group

## Facts used

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