Generalizing Part 1 of the above theorem, we have \(n\Z\leq \Z\) for every \(n\in \Z^+\text{.}\)

The proof of this is left as an exercise for the reader.

Example4.2.4

Consider the group \(\langle F,+\rangle\text{,}\) where \(F\) is the set of all functions from \(\R\) to \(\R\) and \(+\) is pointwise addition. Which of the following are subgroups of \(F\text{?}\)

\(H=\{f\in F: f(5)=0\}\text{;}\)

\(K=\{f\in F: f \text{ is continuous} \}\text{;}\)

\(L=\{f\in F: f \text{ is differentiable} \}\text{.}\)

Are any of \(H\text{,}\) \(K\text{,}\) and \(L\) subgroups of one another?

In fact, we can narrow down the number of facts we need to check to prove a subset \(H\subseteq G\) is a subgroup of \(G\) to only two.

Theorem4.2.5Two-Step Subgroup Test

Let \(G\) be a group and \(H\subseteq G\text{.}\) Then \(H\) is a subgroup of \(G\) if

\(H\neq \emptyset\text{;}\) and

For each \(a,b\in H\text{,}\) \(ab^{-1}\in H\text{.}\)

Assume that the above two properties hold. Since \(H\neq
\emptyset\text{,}\) there exists an \(x\in G\) such that \(x\in H\text{.}\) Then \(e_G=xx^{-1}\) is in \(H\text{,}\) by the second property. Next, for every \(a\in H\) we have \(a^{-1}=e_Ga^{-1}\in H\) (again by the second property). Finally, if \(a,b\in H\) then we've already shown \(b^{-1}\in H\text{;}\) so \(ab=a(b^{-1})^{-1}\in H\text{,}\) yet again by the second property. Thus, \(H\leq G\text{.}\)

Example4.2.6

Use the Two-Step Subgroup Test to prove that \(3\Z\) is a subgroup of \(\Z\text{.}\)

Use the Two-Step Subgroup Test to prove that \(SL(n,\R)\) is a subgroup of \(GL(n,\R)\text{.}\)

It is straightforward to prove the following theorem.

Theorem4.2.7

If \(H\) is a subgroup of a group \(G\) and \(K\) is a subset of \(H\text{,}\) then \(K\) is a subgroup of \(H\) if and only if it's a subgroup of \(G\text{.}\)

It can be useful to look at how subgroups of a group relate to one another. One way of doing this is to consider subgroup lattices (also known as subgroup diagrams). To draw a subgroup lattice for a group \(G\text{,}\) we list all the subgroups of \(G\text{,}\) writing a subgroup \(K\) below a subgroup \(H\text{,}\) and connecting them with a line, if \(K\) is a subgroup of \(H\) and there is no proper subgroup \(L\) of \(H\) that contains \(K\) as a proper subgroup.

Example4.2.8

Consider the group \(\Z_8\text{.}\) We will see later that the subgroups of \(\Z_8\) are \(\{0\}\text{,}\) \(\{0,2,4,6\}\text{,}\) \(\{0,4\}\) and \(\Z_8\) itself. So \(\Z_8\) has the following subgroup lattice.

Example4.2.9

Referring to Example 4.2.4, draw the portion of the subgroup lattice for \(F\) that shows the relationships between itself and its proper subgroups \(H\text{,}\) \(K\text{,}\) and \(L\text{.}\)

Example4.2.10

Indicate the subgroup relationships between the following groups: \(\Z\text{,}\) \(12\Z\text{,}\) \(\Q^+\text{,}\) \(\R\text{,}\) \(6\Z\text{,}\) \(\R^+\text{,}\) \(3\Z\text{,}\) \(G=\langle \{\pi^n:n\in \Z\},\cdot\,\rangle\) and \(J=\langle \{6^n:n\in \Z\},\cdot\,\rangle .\)

We end with a theorem about homomorphisms and subgroups that leads us to another group invariant.

Theorem4.2.11

Let \(G\) and \(G'\) be groups, let \(\phi\) a homomorphism from \(G\) to \(G'\text{,}\) and let \(H\) a subgroup of \(G\text{.}\) Then \(\phi(H)\) is a subgroup of \(G'\text{.}\)

The proof is left as an exercise for the reader.

Corollary4.2.12

If \(G\simeq G'\) and \(G\) contains exactly \(n\) subgroups (\(n\in \Z^+\)), then so does \(G'\text{.}\)

This is another way of, for instance, distinguishing between the groups \(\Z_4\) and the Klein 4-group \(\Z_2^2\text{.}\)

Example4.2.13

By inspection, \(\Z_4\) and \(\Z_2^2\) have, respectively, the following subgroup lattices.

Since \(\Z_4\) contains exactly 3 subgroups and \(\Z_2^2\) exactly 5, we have that \(\Z_4\not\simeq \Z_2^2\text{.}\)