# Nilpotent implies center is normality-large

From Groupprops

This article gives the statement, and possibly proof, of the fact that in any nilpotent group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., normality-large subgroup)

View all such subgroup property satisfactions OR View more information on subgroup-defining functions in nilpotent groups

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., nilpotent group) must also satisfy the second group property (i.e., group whose center is normality-large)

View all group property implications | View all group property non-implications

Get more facts about nilpotent group|Get more facts about group whose center is normality-large

## Contents

## Statement

### Verbal statement

In a nilpotent group, the center is a normality-large subgroup; in other words, the intersection of the center with any nontrivial normal subgroup is a nontrivial normal subgroup.

## Related facts

### Similar facts

### Generalizations

### Applications

- Nilpotent and non-abelian implies center is not complemented
- Minimal normal implies central in nilpotent
- Socle equals Omega-1 of center in nilpotent p-group
- Formula for number of minimal normal subgroups of group of prime power order
- Congruence condition relating number of normal subgroups containing minimal normal subgroups and number of normal subgroups in the whole group
- Thompson's critical subgroup theorem

### Analogues in other algebraic structures

## Proof

**Given**: A nilpotent group with center . A nontrivial normal subgroup of .

**To prove**: is nontrivial.

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Denote by the member of the upper central series of . So is trivial, is the center, and where is the nilpotency class of . | is nilpotent. | |||

2 | There exists some such that is trivial and is nontrivial. | is nontrivial. | Step (1) | [SHOW MORE] | |

3 | . | Definition of upper central series | |||

4 | . | is normal in . | |||

5 | . | By definition of commutator of two subgroups in terms of the commutators being a generating set. | |||

6 | is trivial. | Steps (2),(3),(4),(5) | [SHOW MORE] | ||

7 | , and is nontrivial. | Steps (2), (6) | <toggledisplay>By Step (6), is contained in , so we have a nontrivial subgroup contained in both and . So, is nontrivial. Reviewing the definition of from Step (2), we obtain that must equal 1. |