2-subnormality is not finite-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
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A join of two 2-subnormal subgroups of a group need not be 2-subnormal.
- 2-subnormal implies join-transitively subnormal: The join of a 2-subnormal subgroup and a -subnormal subgroup is -subnormal. In particular, the join of two 2-subnormal subgroups is 4-subnormal, and a join of 2-subnormal subgroups is -subnormal.
- Normal implies join-transitively 2-subnormal, and more generally, join of normal and subnormal implies subnormal of same depth: A join of a normal subgroup and a 2-subnormal subgroup is 2-subnormal. More generally, the join of a normal subgroup and a -subnormal subgroup is -subnormal.
Example of a group of order
We discuss here the example of a group of order 64, given as follows (here denotes the identity element):
The two 2-subnormal subgroups are and , and their join, which is , is 3-subnormal but not 2-subnormal.
Note that the GAP ID of is : it is the among GAP's list of groups of order .
The lattice of the relevant subgroups of is depicted in the diagram below:
Let's go over this step by step:
- The subgroup is a dihedral group, which we call . This has three subgroups of order four: the subgroup , the subgroup , and the subgroup . What happens is that is 3-subnormal, whereas both subgroups and are 2-subnormal.
- is 3-subnormal but not 2-subnormal: In fact, has a subnormal series , where . Here, and . This is both the fastest ascending and the fastest descending subnormal series: the normal closure of in is , and the normal closure of in is . Note that the presentation is given in a way that fits this subnormal series well.
- is 2-subnormal: This needs careful understanding: the normalizer of in is , and this is not normal in . (Thus, is not a 2-hypernormalized subgroup). Nonetheless, is 2-subnormal, because the normal closure of , defined as , is contained in the normalizer.
- is 2-subnormal: The normalizer of in is , while its normal closure is the subgroup . Since the normal closure is contained in the normalizer, is 2-subnormal. Note that in this case, it is also true that is a 2-hypernormalized subgroup: the normalizer of is normal in .