Subnormality satisfies transfer condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., transfer condition)
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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 is a subnormal subgroup of a group , and is a subgroup. Then, is a subnormal subgroup of . Moreover, if is -subnormal in , is -subnormal in (i.e., its subnormal depth is at most .
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
- Normality satisfies transfer condition: If is a normal subgroup of a group , and , then is a normal subgroup of .
- Transfer condition is subordination-closed: If is a subgroup property satisfying the transfer condition, the subordination of .
Given: A group , a -subnormal subgroup of , and a subgroup of .
To prove: is a -subnormal subgroup of .
Proof: Since is -subnormal subgroup of , we have a subnormal series:
We claim that the following is a subnormal series for in :
For this, we need to show that each is normal in . For this, note that:
Since is normal in , fact (1) tells us that is normal in , completing the proof.
This follows directly from facts (1) and (2).