# ACIC is characteristic subgroup-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., ACIC-group) satisfying a group metaproperty (i.e., characteristic subgroup-closed group property)

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## Statement

Suppose is an ACIC-group, i.e., every automorph-conjugate subgroup of is characteristic in (note that characteristic implies automorph-conjugate, so in such a group, the notion of being a characteristic subgroup precisely coincides with the notion of being an automorph-conjugate subgroup).

Suppose is a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of . Then, is also an ACIC-group.

## Generalizations

## Definitions used

Term | Definitions |
---|---|

automorph-conjugate subgroup | A subgroup of a group is termed automorph-conjugate if any subgroup of automorphic to is conjugate to . |

ACIC-group | A group is termed ACIC if every automorph-conjugate subgroup of is characteristic in ; equivalently, every automorph-conjugate subgroup of is normal in . |

## Facts used

## Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

*Given*: An ACIC-group , a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of , and an automorph-conjugate subgroup of .

*To prove*: is normal in , or equivalently, is characteristic in .

*Proof*:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is automorph-conjugate in . | Fact (1) | is automorph-conjugate in , is automorph-conjugate in . | Fact-given direct. | |

2 | is normal in . | is ACIC. | Step (1) | Step-given direct. | |

3 | is normal in . | Fact (2) | . | Step (2) | Step-given-fact direct. |