Groupprops, The Group Properties Wiki (pre-alpha)

YOUR FEEDBACK IS IMPORTANT!

Please take a short user satisfaction survey about Groupprops.

Your survey responses will be helpful in improving the site experience!

Thanks in advance!

From Groupprops

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)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications |Get help on looking up subgroup property implications/non-implications
Get more facts about normal subgroup| Get more facts about characteristic-potentially characteristic subgroup

EXPLORE EXAMPLES YOURSELF: |

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).