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
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 .
- 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).
Given: A nilpotent group , a 2-subnormal subgroup of . has order , with a prime number.
To prove: The normal closure has nilpotency at most
|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]|