# Normal closure of 2-subnormal subgroup of prime power order in nilpotent group has nilpotency class at most equal to prime-base logarithm of order

From Groupprops

## Statement

Suppose is a nilpotent group and is a 2-subnormal subgroup (?) of of order for some prime number . Then, the Normal closure (?) of in is a nilpotent group and its nilpotency class is at most equal to .

Note that since prime power order implies nilpotent, the result applies in particular whenever itself is a finite p-group.

## Related facts

### Similar facts

### Opposite facts

## Facts used

- Normal of prime power order implies contained in upper central series member corresponding to prime-base logarithm of order in nilpotent
- Upper central series members are characteristic
- Characteristic of normal implies normal
- Nilpotency is subgroup-closed (more specifically, nilpotency of fixed class is subgroup-closed).

## Proof

**Given**: A nilpotent group , a 2-subnormal subgroup of . has order , with a prime number.

**To prove**: The normal closure has nilpotency at most

**Proof**:

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

1 | There exists a normal subgroup of such that is a normal subgroup of . | Definition of 2-subnormal | is 2-subnormal in . | ||

2 | is nilpotent. | Fact (4) | is nilpotent. | Step (1) | |

3 | is contained in , i.e., the member of the upper central series of , which is a characteristic subgroup of of nilpotency class at most . | Facts (1), (2) | has order | Steps (1), (2) | Fact+Given+Step direct. |

4 | is a normal subgroup of of nilpotency class at most . | Fact (3) | Steps (1), (3) | [SHOW MORE] | |

5 | The normal closure is contained in . | Steps (3), (4) | [SHOW MORE] | ||

6 | has nilpotency class at most . | Steps (4), (5) | [SHOW MORE] |