Template:Groups of prime order
This article gives information about, and links to more details on, groups of order {{{1}}}
See pages on algebraic structures of order {{{1}}}"{{{" can not be assigned to a declared number type with value 1. | See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order {{{1}}}. See also more detailed information on specific subtopics through the links:
| Information type | Page summarizing information for groups of order {{{1}}} |
|---|---|
| element structure (element orders, conjugacy classes, etc.) | [[element structure of groups of order {{{1}}}]] |
| subgroup structure | [[subgroup structure of groups of order {{{1}}}]] |
| linear representation theory | [[linear representation theory of groups of order {{{1}}}]] [[projective representation theory of groups of order {{{1}}}]] [[modular representation theory of groups of order {{{1}}}]] |
| endomorphism structure, automorphism structure | [[endomorphism structure of groups of order {{{1}}}]] |
| group cohomology | [[group cohomology of groups of order {{{1}}}]] |
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 {{{2}}}. 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 {{{2}}}, namely [[cyclic group:Z{{{1}}}]]. This follows from the fact that {{{2}}} 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 {{{2}}} are non-trivial abelian groups, they are certainly all nilpotent groups, of nilpotency class . Alternatively, since {{{2}}} is prime and hence is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.
