# Slender nilpotent and every proper subgroup is abelian implies Frattini-in-center

From Groupprops

## Contents

## Statement

Suppose is a Slender nilpotent group (?), i.e., a group that is both a Nilpotent group (?) and a Slender group (?) (every subgroup is finitely generated, or equivalently, every ascending chain of subgroups stabilizes after a finite length). Then, is a Frattini-in-center group (?), i.e., the commutator subgroup of is contained in the Frattini subgroup of , which in turn is contained in the center of .

## Related facts

- Finite non-abelian and every proper subgroup is abelian implies not simple
- Finite non-abelian and every proper subgroup is abelian implies metabelian
- Classification of finite non-abelian groups in which every proper subgroup is abelian
- Schmidt-Iwasawa theorem

## Facts used

## Proof

**Given**: A slender nilpotent group such that every proper subgroup of is abelian.

**To prove**: .

**Proof**: If is abelian, we are done, so we assume that is non-abelian.

- Every non-identity element of is contained in a maximal subgroup of : [SHOW MORE]
- has at least two maximal subgroups and , both of which are normal in it: [SHOW MORE]
- : [SHOW MORE]
- is contained in the center of : [SHOW MORE]
- : [SHOW MORE]
- (
**Given data used**: nilpotent) : [SHOW MORE]

The last two steps complete the proof.