Finite group having a subgroup series with prime indexes
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
A finite group having a subgroup series with prime indices is a finite group with a subgroup series starting at the trivial subgroup and terminating at the whole group, where each subgroup has prime index in its successor.
In other words, it is a finite group whose max-length equals the sum of exponents of its prime divisors in the prime factorization of its order.
The smallest finite group that does not have this property is the alternating group of degree six.
Relation with other properties
- Finite cyclic group
- Finite abelian group
- Finite nilpotent group
- Finite supersolvable group
- Finite solvable group
This group property is normal subgroup-closed: any normal subgroup (and hence, any subnormal subgroup) of a group with the property, also has the property
View other normal subgroup-closed group properties
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties
This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties