Subgroup structure of groups of order 7
This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 7.
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This article gives information on the subgroup structure of groups of order 7. There is one group of this order, namely cyclic group:Z7, since 7 is a prime number. (More generally, a cyclicity-forcing number.)
By Lagrange's theorem, the order of subgroups of a group of order 7 must be 1 or 7. Also, the subgroup of a cyclic group is cyclic.
Thus, the only subgroups of cyclic group:Z7 are cyclic group:Z7 and trivial group.
Since cyclic group:Z7 is abelian, these are normal subgroups. Thus, cyclic group:Z7 is a simple group.
Subgroup-defining functions
| Subgroup-defining function | What it means | Value as subgroup for cyclic group:Z7 |
|---|---|---|
| center | elements that commute with every group element | whole group (cyclic group:Z7) |