Template:Subgroup structure of groups of prime order

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order {{{1}}}.
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This article gives information on the subgroup structure of groups of order {{{1}}}. There is one group of this order, namely [[cyclic group:Z{{{1}}}]], since {{{1}}} is a prime number. (More generally, a cyclicity-forcing number.)

By Lagrange's theorem, the order of subgroups of a group of order {{{1}}} must be 1 or {{{1}}}. Also, the subgroup of a cyclic group is cyclic.

Thus, the only subgroups of [[cyclic group:Z{{{1}}}]] are [[cyclic group:Z{{{1}}}]] and trivial group.

Since [[cyclic group:Z{{{1}}}]] is abelian, these are normal subgroups. Thus, [[cyclic group:Z{{{1}}}]] is a simple group.

Subgroup-defining functions

Subgroup-defining function	What it means	Value as subgroup for [[cyclic group:Z{{{1}}}]]
center	elements that commute with every group element	whole group ([[cyclic group:Z{{{1}}}]])