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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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.
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 about normality and subnormality
- Normality is strongly UL-intersection-closed
- Normality satisfies transfer condition
- Normality satisfies inverse image condition
- Normality satisfies intermediate subgroup condition
- Normality satisfies transfer condition: If are subgroups such that is normal in , then is normal in .
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.