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.

  • 1
    $\begingroup$ $\{-1,0,1\}$ is not a subgroup of $\Bbb Z$ $\endgroup$ – janmarqz Oct 5 '15 at 23:23
  • $\begingroup$ Do you go to uic? $\endgroup$ – Yunus Syed Oct 5 '15 at 23:54
  • $\begingroup$ @janmarqz Ahh... I see, stupid error on my part. Thank you. $\endgroup$ – user1105224 Oct 6 '15 at 1:04
  • $\begingroup$ @YunusSyed No, I'm learning this on my own time. $\endgroup$ – user1105224 Oct 6 '15 at 1:05
  • $\begingroup$ Oh, I thought you were in my algebra class. $\endgroup$ – Yunus Syed Oct 6 '15 at 1:17

As pointed out in the other answer, a subgroup needs to be closed under the operation, so your examples are not subgroups (eg $3+3=6$ is not in the set). There is always a one element subgroup consisting only of the identity, but that subgroup isn't really interesting. It's called the trivial subgroup, and since it is always the smallest, when we speak of minimal subgroups we are talking about minimal nontrivial subgroups. Similarly, the entire group is not generally considered a maximal subgroup.


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