Vipul: New page: {{subgroup metaproperty satisfaction| property = subnormal subgroup| metaproperty = transfer condition}} ==Statement== ===Verbal statement=== The intersection of a [[subnormal subgroup]...

2008-10-20T22:39:42Z

New page: {{subgroup metaproperty satisfaction| property = subnormal subgroup| metaproperty = transfer condition}} ==Statement== ===Verbal statement=== The intersection of a [[subnormal subgroup]...

New page

{{subgroup metaproperty satisfaction|
property = subnormal subgroup|
metaproperty = transfer condition}}

==Statement==

===Verbal statement===

The intersection of a [[subnormal subgroup]] with any subgroup is subnormal in that subgroup. Moreover, the [[subnormal depth]] of the intersection is bounded from above by the subnormal depth of the original subgroup.

===Statement with symbols===

Suppose <math>H</math> is a subnormal subgroup of a group <math>G</math>, and <math>K \le G</math> is a subgroup. Then, <math>H \cap K</math> is a subnormal subgroup of <math>K</math>. Moreover, if <math>H</math> is <math>k</math>-subnormal in <math>G</math>, <math>H \cap K</math> is <math>k</math>-subnormal in <math>K</math> (i.e., its subnormal depth is at most <math>k</math>.

==Related facts==

===Related facts about subnormality===

* [[Subnormality satisfies intermediate subgroup condition]]
* [[Subnormality satisfies inverse image condition]]
* [[Subnormality satisfies image condition]]

===Related facts about normality===

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

==Facts used==

# [[uses::Normality satisfies transfer condition]]: If <math>H</math> is a normal subgroup of a group <math>G</math>, and <math>K \le G</math>, then <math>H \cap K</math> is a normal subgroup of <math>K</math>.
# [[uses::Transfer condition is subordination-closed]]: If <math>p</math> is a subgroup property satisfying the transfer condition, the [[subordination operator|subordination]] of <math>p</math>.

==Proof==

===Hands-on proof===

'''Given''': A group <math>G</math>, a <math>k</math>-subnormal subgroup <math>H</math> of <math>G</math>, and a subgroup <math>K</math> of <math>G</math>.

'''To prove''': <math>H \cap K</math> is a <math>k</math>-subnormal subgroup of <math>K</math>.

'''Proof''': Since <math>H</math> is <math>k</math>-subnormal subgroup of <math>G</math>, we have a subnormal series:

<math>H = H_0 \le H_1 \le \dots \le H_k = G</math>.

We claim that the following is a subnormal series for <math>H \cap K</math> in <math>K</math>:

<math>H \cap K = H_0 \cap K \le H_1 \cap K \le \dots H_k \cap K = K</math>.

For this, we need to show that each <math>H_i \cap K</math> is normal in <math>H_{i+1} \cap K</math>. For this, note that:

<math>H_i \cap K = H_i \cap (H_{i+1} \cap K)</math>.

Since <math>H_i</math> is normal in <math>H_{i+1}</math>, fact (1) tells us that <math>H_i \cap (H_{i+1} \cap K) = H_i \cap K</math> is normal in <math>H_{i+1} \cap K</math>, completing the proof.

===Property-theoretic proof===

This follows directly from facts (1) and (2).

Subnormality satisfies transfer condition - Revision history

Vipul: New page: {{subgroup metaproperty satisfaction| property = subnormal subgroup| metaproperty = transfer condition}} ==Statement== ===Verbal statement=== The intersection of a [[subnormal subgroup]...