# Section4.2Subgroup proofs and lattices¶ permalink

Using Lemma 4.1.6 and the argument preceding it, we have the following.

##### Example4.2.2

For each of the following, prove that the given subset $H$ of group $G$ is or is not a subgroup of $G\text{.}$

1. $H=3\Z\text{,}$ $G=\Z\text{.}$

2. $H=\{0,1,2,3\}\text{,}$ $G=\Z_6\text{;}$

3. $H=\R^*\text{,}$ $G=\R\text{;}$

4. $H=\{(0,x,y,z):x,y,z\in \R\}\text{,}$ $G=\R^4\text{.}$

##### Example4.2.3

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 $F\text{?}$

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

2. $K=\{f\in F: f \text{ is continuous} \}\text{;}$

3. $L=\{f\in F: f \text{ is differentiable} \}\text{.}$

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.

##### Example4.2.6

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

2. 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.

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.

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 $\Z_4\not\simeq \Z_2^2\text{.}$