# Orbital maximin problem

From Groupprops

## Statement

For a given natural number and a group property , the **orbital maximin problem** for the pair is the problem of finding a group satisfying with an action on a set of size such that the minimum of the sizes of the orbitals under the action of is as large as possible.

Here, an orbital is an orbit under the induced action of on unordered pairs of distinct elements from the set.

In general, we take to be a property that is both subgroup-closed and quotient-closed. For such , we can restrict attention to faithful group actions.

This is related to the orbital maximax problem, where we try to maximize the size of the largest orbital.

## Particular cases

### Exact values and conditional exact values

Group property/constraint on action | Value for orbital maximin | Proof |
---|---|---|

cyclic group | follows from orbital maximin equals size of set for abelian groups | |

abelian group | orbital maximin equals size of set for abelian groups | |

nilpotent group | orbital maximin equals size of set for nilpotent groups | |

Group with nontrivial center acting faithfully | orbital maximin equals size of set for faithful actions by groups with nontrivial center | |

solvable group | iff is a prime power | orbital maximin equals number of ordered pairs of distinct elements for solvable groups iff size is prime power |

group satisfying Oliver's condition | iff is a prime power | (see above) |

metabelian group | iff is a prime power | (see above) |

### Bounds

Group property/constraint on action | Bound for orbital maximin | Proof/discussion |
---|---|---|

solvable group | bounded below by a constant times | orbital maximin is bounded below by constant fraction of number of ordered pairs of distinct elements for solvable groups |

metabelian group | bounded below by a constant times | (see above) |

group satisfying Oliver's condition | bounded below at least by , probably | orbital maximin problem for groups satisfying Oliver's condition |