Subgroup structure of groups of order 3

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 3.
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This article gives information on the subgroup structure of groups of order 3. There is one group of this order, namely cyclic group:Z3, since 3 is a prime number. (More generally, a cyclicity-forcing number.)

By Lagrange's theorem, the order of subgroups of a group of order 3 must be 1 or 3. Also, the subgroup of a cyclic group is cyclic.

Thus, the only subgroups of cyclic group:Z3 are cyclic group:Z3 and trivial group.

Since cyclic group:Z3 is abelian, these are normal subgroups. Thus, cyclic group:Z3 is a simple group.

Subgroup-defining functions

Subgroup-defining function	What it means	Value as subgroup for cyclic group:Z3
center	elements that commute with every group element	whole group (cyclic group:Z3)