Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?
Am I correct in thinking that it would:
1) Identity element exists by assumption
2) All g in G have an inverse (the identity is its own inverse)
3) composition under the operation of the identity is associative
A subgroup consisting only of the identity is, along with the "subgroup" created by all elements of the group itself, is called a "trivial subgroup." Yes, it is a subgroup.
A "proper subgroup" is a subgroup that is not just the identity, nor the entire group.
it is quite common yes. for example, excluding the trivial subgroup as a proper subgroup allows the characterization of a simple group as a group with no proper normal subgroups, rather than a group with no nontrivial proper normal subgroups. however, this can vary from author to author.