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

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 2) 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 2

n 2^n List of groups of order 2^n 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.

Relation with other properties

Stronger properties