# Finite group in which half or more of the elements are involutions

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **finite group in which half or more of the elements are involutions** is a finite group in which the number of involutions (i.e., elements of order equal to ) is half or more of the order of the group.

This property is determined completely by looking at the order statistics of the group.

## Examples

### Groups of order powers of

List of groups of order in which half or more of the elements are involutions | List of GAP IDs (in same order) | Those groups in this list that are not generalized dihedral groups |
List of GAP IDs (in same order) | ||
---|---|---|---|---|---|

1 | 2 | cyclic group:Z2 | 1 | -- | |

2 | 4 | Klein four-group | 2 | -- | |

3 | 8 | dihedral group:D8, elementary abelian group:E8 | 3, 5 | -- | |

4 | 16 | dihedral group:D16, direct product of D8 and Z2, elementary abelian group:E16 | 7, 11, 14 | -- | |

5 | 32 | dihedral group:D32, SmallGroup(32,27), generalized dihedral group for direct product of Z4 and Z4, direct product of D16 and Z2, direct product of D8 and V4, inner holomorph of D8, elementary abelian group:E32 | 18, 27, 34, 39, 45, 49, 51 | SmallGroup(32,27), inner holomorph of D8 | 27, 49 |

6 | 64 | (too long to list) | 52, 174, 186, 202, 211, 226, 250, 261, 264, 267 | direct product of SmallGroup(32,27) and Z2, direct product of D8 and D8, direct product of SmallGroup(32,49) and Z2 | 202, 226, 264 |

### Groups of other orders

For all other orders, the only examples are the generalized dihedral groups.