# Frattini-embedded normal in subgroup and normal implies Frattini-embedded normal

## Contents

## Statement

Suppose are groups, such that:

- is a normal subgroup of
- is a Frattini-embedded normal subgroup of . In other words, is normal in and for any proper subgroup of , is a proper subgroup

Then is a Frattini-embedded normal subgroup of .

## Related facts

- Frattini subgroup is normal-monotone: The Frattini subgroup of a normal subgroup is contained in the Frattini subgroup of the whole group, when the group satisfies the property that every proper subgroup is contained in a maximal subgroup. The proof of this monotonicity uses the statement of this article, along with the fact that a characteristic subgroup of a normal subgroup must be normal.

## Definitions used

### Frattini-embeddded normal subgroup

`Further information: Frattini-embedded normal subgroup`

A subgroup of a group is termed **Frattini-embedded normal** in if and, for any proper subgroup of , is proper.

## Proof

*Given*: are groups, such that:

- is a normal subgroup of
- is a Frattini-embedded normal subgroup of . In other words, is normal in and for any proper subgroup of , is a proper subgroup

*To prove*: For any proper subgroup of , is a proper subgroup of

*Proof*: Pick a proper subgroup of . We need to show that is proper in . Suppose . We'll derive a contradiction.

We have , which, by the modular property of groups, gives:

Now, clearly, is not contained in (otherwise ), and since , is not contained in . So is a proper subgroup of , and thus we have found a proper subgroup of whose product with equals . This contradicts the assumption that is a Frattini-embedded normal subgroup of .