I am working through "Abstract Algebra:Theory and Applications" by Thomas W. Judson (http://abstract.ups.edu/aata/index.html). I have a question related to the definition of groups, specifically in relation to the size of a subgroup of a group. The text presents a couple of questions and a theorem related to the size of a subgroup which makes me think I am not understanding the definition of subgroup correctly.
The definition of subgroups given in the text:
Sometimes we wish to investigate smaller groups sitting inside a larger group. The set of even integers 2Z={…,−2,0,2,4,…} is a group under the operation of addition. This smaller group sits naturally inside of the group of integers under addition. We define a subgroup H of a group G to be a subset H of G such that when the group operation of G is restricted to H, H is a group in its own right. Observe that every group G with at least two elements will always have at least two subgroups, the subgroup consisting of the identity element alone and the entire group itself. The subgroup H={e} of a group G is called the trivial subgroup. A subgroup that is a proper subset of G is called a proper subgroup. In many of the examples that we have investigated up to this point, there exist other subgroups besides the trivial and improper subgroups.
Questions and theorem confusing me:
Let n=0,1,2,… and nZ={nk:k∈Z}. Prove that nZ is a subgroup of Z. Show that these subgroups are the only subgroups of Z.
Give an example of an infinite group in which every nontrivial subgroup is infinite.
Let G be a group and a be any element in G. Then the set ⟨a⟩={ak:k∈Z} is a subgroup of G. Furthermore, ⟨a⟩ is the smallest subgroup of G that contains a.
My source of confusion:
According to the definition in the text (which seems to match other definitions I've found by Googling), I am under the impression that any subset, H, of a group, G, should be considered a subgroup of G so long as the elements of H form a group under the operation inherited by G.
However, this seems to contradict the above questions and theorem. A group, defined as I understand it, should only need an identity element, closure, and an inverse element for each element in its set. To me, this seems to imply that every non-trivial group has a smallest non-trivial subgroup composed of a maximum of 3 elements, and a minimum of 2 elements: the identity element, e, any other element, a, and the inverse of a (where in some cases a is its own inverse). For example, {-1, 0, 1}, {-2, 0, 2}, {-3, 0, 3}... could be subgroups of the integers.
Excluding these 2-3 element subgroups I understand the above theorem and can answer the above questions. Am I misunderstanding something here? Or does my text possibly have some "loose" definitions that are leading me astray? Any help resolving this confusion would be appreciated.
