# Subnormality of fixed depth satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)

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## Contents

## Statement

### Verbal statement

A subnormal subgroup of a group is also subnormal in every intermediate subgroup. In fact, its subnormal depth in any intermediate subgroup is bounded from above by the subnormal depth in the whole group.

### Property-theoretic statement

The subgroup property of being a subnormal subgroup satisfies the subgroup metaproperty called the intermediate subgroup condition -- any subnormal subgroup of the whole group is also subnormal in every intermediate subgroup.

### Statement with symbols

Suppose is a subnormal subgroup of a group . Then, for any intermediate subgroup (i.e., ), is subnormal in . Moreover, if is -subnormal in , is also -subnormal in . (Here, when we say -subnormal, we mean the subnormal depth is *at most* ).

## Related facts

### Generalizations

### Related facts about normality and subnormality

- Normality is strongly UL-intersection-closed
- Normality satisfies transfer condition
- Normality satisfies inverse image condition
- Normality satisfies intermediate subgroup condition

## Facts used

- Normality satisfies transfer condition: If are subgroups such that is normal in , then is normal in .

## Proof

### Hands-on proof

**Given**: A group , a -subnormal subgroup , a subgroup such that .

**To prove**: is -subnormal in .

**Proof**: Consider a subnormal series for of length :

.

where is normal in for each . We claim that the series:

is a subnormal series for in . For this, observe that:

.

We know that is normal in , so by fact (1), is normal in , yielding that is normal in , as desired.