Subnormality is permuting upper join-closed in finite

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This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup of finite group) satisfying a subgroup metaproperty (i.e., permuting upper join-closed subgroup property)
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History

The result was proved both by Maier and by Wielandt.

Statement

When the whole group is finite

Suppose G is a finite group and H is a subgroup of G. Suppose K_1, K_2 are intermediate subgroups of G such that K_1K_2 = K_2K_1 (i.e., they are Permuting subgroups (?)) and H is a subnormal subgroup in both K_1 and K_2. Then, H is also subnormal in the product of subgroups K_1K_2.

Equivalent formulation when the whole group is not finite

Suppose G is a group and H is a finite subgroup of G. Suppose K_1, K_2 are intermediate finite subgroups of G such that K_1K_2 = K_2K_1 (i.e., they are Permuting subgroups (?)) and H is a subnormal subgroup in both K_1 and K_2. Then, H is also subnormal in the product of subgroups K_1K_2.

Note that these two formulations are equivalent because even if G is not finite, the product K_1K_2 is still finite since both K_1 and K_2 are finite.

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