Groups of order 89
This article gives information about, and links to more details on, groups of order 89
See pages on algebraic structures of order 89 | See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order 89. See also more detailed information on specific subtopics through the links:
| Information type | Page summarizing information for groups of order 89 |
|---|---|
| element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 89 |
| subgroup structure | subgroup structure of groups of order 89 |
| linear representation theory | linear representation theory of groups of order 89 projective representation theory of groups of order 89 modular representation theory of groups of order 89 |
| endomorphism structure, automorphism structure | endomorphism structure of groups of order 89 |
| group cohomology | group cohomology of groups of order 89 |
Statistics at a glance
| Quantity | Value | Explanation |
|---|---|---|
| Total number of groups | 1 | See classification of groups of prime order |
| Number of abelian groups | 1 | equals the number of unordered integer partitions of 1, the exponent part in the prime factorisation of . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |
| Number of simple groups | 1 | |
| Number of nilpotent groups | 1 | See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. |
There is, up to isomorphism, a unique group of order , namely cyclic group:Z89. This follows from the fact that is prime and there is a unique isomorphism class of group of prime order, namely that of the cyclic group of prime order.
Since all the groups of order are non-trivial abelian groups, they are certainly all nilpotent groups, of nilpotency class . Alternatively, since is prime and hence is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.
