# Finite non-nilpotent and every proper subgroup is nilpotent implies not simple

From Groupprops

## Statement

Suppose is a finite group that is *not* a nilpotent group, but every proper subgroup of is a (and in particular, a ). Then, cannot be a simple group.

## Facts used

- Nilpotent implies normalizer condition: In a nilpotent group, every proper subgroup is properly contained in its normalizer.
- Finite and any two maximal subgroups intersect trivially implies not simple non-abelian

## Proof

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**Given**: A finite non-nilpotent group such that every proper subgroup of is nilpotent.

**To prove**: is not simple.

**Proof**: We assume that is simple, and derive a contradiction. Let be the number of elements of .

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

1 | The intersection of any two maximal subgroups of is contained in an intersection of two maximal subgroups that is a normal subgroup of . | Fact (1) | Every proper subgroup of is nilpotent. is finite |
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2 | Any two maximal subgroups of intersect trivially. | is simple | Step (1) | [SHOW MORE] | |

3 | We have the desired contradiction. | Fact (2) | is simple non-nilpotent, hence simple non-abelian | Step (2) | Follows directly from Fact (2). |