# Base of a wreath product implies right-transitively 2-subnormal

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., 2-subnormal subgroup (?) and Base of a wreath product (?)), to another known subgroup property (i.e., 2-subnormal subgroup (?))
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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., base of a wreath product) must also satisfy the second subgroup property (i.e., right-transitively 2-subnormal subgroup)
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## Statement

### Property-theoretic statement

The statement has the following equivalent forms:

1. The subgroup property of being the base of a wreath product is stronger than the subgroup property of being a right-transitively 2-subnormal subgroup.
2. The composition of the subgroup property of being 2-subnormal with the subgroup property of being the base of a wreath product implies the subgroup property of being 2-subnormal.

2-subnormal  Base of a wreath product  2-subnormal

### Verbal statement

1. Any base of a wreath product in a group is a right-transitively 2-subnormal subgroup.
2. Any 2-subnormal subgroup of the base of a wreath product is 2-subnormal in the whole group.

### Statement with symbols

1. If  is the base of a wreath product in , then  is a right-transitively 2-subnormal subgroup of .
2. If  is a 2-subnormal subgroup of  and  is the base of a wreath product in , then  is 2-subnormal in .

## Facts used

1. Direct factor implies transitively normal: Any normal subgroup of a direct factor is normal.

## Proof

Given: A group  with  the base of a wreath product. In other words,  (the direct product may be infinite), and the subgroup  is the first direct factor.  is a 2-subnormal subgroup of . Let .

To prove:  is 2-subnormal in .

Proof: Let  be the normal closure of  in .

1. (Fact used: Fact (1), direct factor implies transitively normal):  is normal in , and  is a direct factor of . Thus,  is normal in .
2. (Given data used:  is a wreath product of  and ): Since  is normal in , the normal closure  of  in  is given as the closure of  under the action of . This is a direct product of isomorphic copies of  for all the coordinates in the orbit of the first coordinate.
3. (Given data used:  is 2-subnormal in ): Since  is 2-subnormal in ,  is normal in its normal closure  in .
4. (Fact used: Fact (1), direct factor implies transitively normal):  is normal in , and  is a direct factor of , so  is normal in .
5. Since  is normal in , and  is normal in  ( is, after all, defined as the normal closure of  in ),  is 2-subnormal in .