Cyclicity is subgroup-closed
This article gives the statement, and possibly proof, of a group property (i.e., cyclic group) satisfying a group metaproperty (i.e., subgroup-closed group property)
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Contents
Statement
Verbal statement
Every subgroup of a cyclic group is cyclic.
Facts used
Proof
For the infinite cyclic group
Any infinite cyclic group is isomorphic to the group of integers , so we prove the result for
. The result follows from fact (1). Note that the trivial subgroup is cyclic anyway, and fact (1) states that every nontrivial subgroup is cyclic on its smallest element.
For a finite cyclic group
Any finite cyclic group is isomorphic to the group of integers modulo n, so it suffices to prove the result for those groups.
Suppose is the group of integers modulo n, and suppose
is a subgroup of
. Define
as the subset of
comprising those elements of
in the congruence classes of
. In other words:
is in the congruence class
Then, is clearly a subgroup of
, because congruences mod
preserve addition, additive inverses and identity elements.
By fact (1), there exists a such that
. Clearly,
, so
. Going back, we see that
is cyclic on the congruence class of
.