Groupprops, The Group Properties Wiki (pre-alpha)

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., characteristic-potentially characteristic subgroup)
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Contents 1 Statement 1.1 Verbal statement 1.2 Statement with symbols 2 Facts used 3 Proof

Statement

Verbal statement

A normal subgroup need not be characteristic-potentially characteristic.

Statement with symbols

It is possible to have a group K and a normal subgroup H of K such that there is no group G containing K in which both H and K are characteristic subgroups.

Facts used

1. Characteristic-potentially characteristic implies normal-potentially characteristic
2. Normal not implies normal-potentially characteristic

Proof

The proof follows directly from facts (1) and (2).