In-normalizer operator

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This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property


View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

Definition

The in-normalizer operator is a map from the subgroup property space to itself that takes as input a subgroup property p and outputs a subgroup property q defined as follows: H satisfies property q in G if and only if H satisfies property p in N_G(H).

Application

Important instances of application of the in-normalizer operator:

Properties

Monotonicity

This subgroup property modifier is monotone, viz if p \le q are subgroup properties and f is the operator, then f(p) \le f(q)

If p \le q the the in-normalizer of p is also \le the in-normalizer of q.

Idempotence

This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once

The in-normalizer operator is idempotent, in the sense that applying it twice to a given subgroup property has the same effect as applying it once. An element is a fixed-point under this operator if and only if it is a in-normalizer subgroup property.

Conditionally ascendant

If p is stronger than the property of being a normal subgroup, then p is stronger than the in-normalizer of p. In any case, the conjunction of p with the property of being normal, is stronger than the in-normalizer of p.

This is true of all the examples mentioned above.

Identity-true implies self-normalizing

The property of being the improper subgroup (that is, being the whole group) gets mapped under the in-normalizer operator to the property of being a self-normalizing subgroup. This, along with the monotonicity, tells us that any identity-true subgroup rpoperty gets mapped under the in-normalizer operator to a property weaker than the property of being a self-normalizing subgroup.