Subgroup Tests

students, imagine a math club where only certain students can join if they follow the club rules. In abstract algebra, a subgroup is like a smaller group hidden inside a bigger one. The big question is: how can we check efficiently whether a set is really a subgroup? 🤔 This lesson explains the main subgroup tests, how to use them, and why they matter for later ideas like generated subgroups and cyclic groups.

By the end of this lesson, you should be able to:

explain what a subgroup test is and why it works,
check whether a subset is a subgroup using the standard tests,
connect subgroup tests to the larger study of groups,
use examples to justify subgroup decisions clearly.

What a subgroup is and why tests matter

A group is a set with an operation that satisfies four rules: closure, associativity, an identity element, and inverses. A subgroup is a subset of a group that is itself a group using the same operation.

For example, the integers $\mathbb{Z}$ under addition form a group. The even integers $2\mathbb{Z} = \{\dots,-4,-2,0,2,4,\dots\}$ are a subgroup of $\mathbb{Z}$ because adding two even integers gives an even integer, $0$ is included, and the negative of an even integer is even.

Why do we need tests? Because checking all group axioms directly every time can be slow. Subgroup tests give shorter methods. Instead of checking everything from scratch, we can use a few key conditions to decide if a subset is a subgroup. This is especially useful in examples with matrices, integers modulo $n$, and symmetry groups.

The basic subgroup test

The most common idea is this:

A nonempty subset $H$ of a group $G$ is a subgroup if $H$ is closed under the group operation and inverses.

For a group written multiplicatively, the test is:

$H \neq \varnothing$,
if $a,b \in H$, then $ab \in H$,
if $a \in H$, then $a^{-1} \in H$.

If these are true, then $H$ is a subgroup of $G$.

Why nonempty matters

If a set is empty, it cannot be a subgroup because a group must contain an identity element. So the first step is always to make sure the set is not empty.

Example: even integers

Let $G = (\mathbb{Z}, +)$ and let $H = 2\mathbb{Z}$.

$H$ is nonempty because $0 \in H$.
If $a, b \in H$, then $a = 2m$ and $b = 2n$ for some integers $m,n$.
Then $a+b = 2m + 2n = 2(m+n) \in H$.
If $a \in H$, then $a = 2m$, so $-a = -2m = 2(-m) \in H$.

So $H$ is a subgroup of $\mathbb{Z}$.

Example: positive integers are not a subgroup

Let $H = \{1,2,3,\dots\}$ inside $\mathbb{Z}$ under addition.

It is nonempty.
It is closed under addition.
But it is not closed under inverses, because the inverse of $1$ is $-1$, and $-1 \notin H$.

So $H$ is not a subgroup.

This example is important because it shows that closure under the operation alone is not enough. A subgroup must also contain inverses.

The one-step subgroup test

There is a very efficient version called the one-step subgroup test.

A nonempty subset $H$ of a group $G$ is a subgroup if for every $a,b \in H$, the element $ab^{-1}$ is also in $H$.

This works for multiplicative notation. For additive groups, the test becomes:

a nonempty subset $H$ of $(G,+)$ is a subgroup if for all $a,b \in H$, the element $a-b$ is in $H$.

This test is powerful because one condition can replace separate closure and inverse checks.

Why this works

If $H$ is a subgroup, then for $a,b \in H$, since $b^{-1} \in H$ and $ab^{-1} \in H$, the condition is automatic.

The deeper direction is: if $H$ is nonempty and closed under $ab^{-1}$, then $H$ contains the identity, inverses, and is closed under the operation.

For example, choose any $a \in H$. Since $H$ is nonempty, there is some $b \in H$. Then $ab^{-1} \in H$. If you let $a=b$, you get $aa^{-1}=e \in H$, so the identity is in $H$.

Once $e \in H$, taking $a=e$ and $b \in H$ gives $eb^{-1}=b^{-1} \in H$, so inverses are included.

Finally, if $a,c \in H$, then because $c^{-1} \in H$, applying the test to $a$ and $c^{-1}$ gives $a(c^{-1})^{-1} = ac \in H$.

So the single condition is enough.

Example: a subset of integers modulo $6$

Consider $G = \mathbb{Z}_6 = \{0,1,2,3,4,5\}$ under addition mod $6$, and let

$$H = \{0,3\}.$$

Check the additive version of the test: for any $a,b \in H$, is $a-b \in H$?

$0-0 = 0 \in H$
$0-3 = -3 \equiv 3 \pmod{6}$, so $3 \in H$
$3-0 = 3 \in H$
$3-3 = 0 \in H$

Yes, so $H$ is a subgroup of $\mathbb{Z}_6$.

How to use subgroup tests in practice

When you are given a subset, follow a simple process:

Check whether the subset is nonempty.
Decide which test is easiest: the full closure-and-inverse test or the one-step test.
Use algebraic form, not just list checking, whenever possible.
Write clear justification for each step.

Example: a subset of real numbers under addition

Let $G = (\mathbb{R}, +)$ and let

$$H = \{x \in \mathbb{R} : x > 0\}.$$

This set is nonempty and closed under addition. But it is not a subgroup because $x > 0$ implies $-x < 0$, so the inverse of an element is not in $H$. Therefore $H$ fails the subgroup test.

Example: multiples of $5$

Let $H = \{5k : k \in \mathbb{Z}\}$ inside $\mathbb{Z}$.

Using the additive one-step test, take $a=5m$ and $b=5n$.

Then

$$a-b = 5m - 5n = 5(m-n),$$

which is again a multiple of $5$. So $H$ is a subgroup.

This is a common pattern: sets defined by divisibility or linear conditions often produce subgroups.

Common mistakes and how to avoid them

A frequent mistake is checking only closure under the operation. students, remember that a subgroup must also include inverses. For example, the set of natural numbers is closed under addition, but it is not a subgroup of the integers because it does not contain additive inverses.

Another mistake is assuming that the identity automatically belongs to any nonempty subset. That is not true. You must prove it or use a subgroup test that guarantees it.

A third mistake is using the wrong operation. The same set can behave differently depending on the group operation. For example, positive real numbers are a subgroup of $(\mathbb{R}_{>0}, \times)$, because multiplication has inverses for positive numbers, but they are not a subgroup of $(\mathbb{R}, +)$.

So always identify:

the parent group,
the subset,
the operation,
and the correct subgroup test.

Connection to generated subgroups and cyclic groups

Subgroup tests are the doorway to later topics in the chapter. A generated subgroup is the smallest subgroup containing a given set of elements. To prove something is a subgroup, you often use a subgroup test first.

For example, in a group $G$, the subgroup generated by an element $a$ is written $\langle a \rangle$. It consists of all powers of $a$ in a multiplicative group, or all integer multiples of $a$ in an additive group. Showing that $\langle a \rangle$ is a subgroup uses subgroup reasoning.

This leads to cyclic groups, which are groups generated by one element. A group is cyclic if it can be written as $\langle a \rangle$ for some $a$. Subgroup tests help you recognize and verify these structures.

For example, $\mathbb{Z}$ is cyclic because $\mathbb{Z} = \langle 1 \rangle$. The subgroup $2\mathbb{Z}$ is itself cyclic because $2\mathbb{Z} = \langle 2 \rangle$.

This is why subgroup tests are not just a technical skill. They are a foundation for understanding how groups are built from smaller pieces.

Conclusion

Subgroup tests give fast, reliable ways to decide whether a subset is a subgroup. The standard test checks that the subset is nonempty, closed under the operation, and closed under inverses. The one-step test simplifies this further by checking whether $ab^{-1}$ stays inside the set, or $a-b$ in additive notation.

These tests are essential in abstract algebra because they help you move from examples to structure. They also prepare you for generated subgroups and cyclic groups, where the idea of building a group from a small set of elements becomes central. students, if you can use subgroup tests confidently, you have a strong tool for reading and proving many results in group theory.

Study Notes

A subgroup is a subset of a group that is itself a group under the same operation.
A nonempty subset $H$ of a group $G$ is a subgroup if it is closed under the operation and inverses.
The one-step subgroup test says: if $H \neq \varnothing$ and $ab^{-1} \in H$ for all $a,b \in H$, then $H$ is a subgroup.
In additive notation, the one-step test becomes: if $H \neq \varnothing$ and $a-b \in H$ for all $a,b \in H$, then $H$ is a subgroup.
Always check the correct operation and the parent group before testing a subset.
Positive integers are not a subgroup of $\mathbb{Z}$ under addition because inverses are missing.
Even integers $2\mathbb{Z}$ are a subgroup of $\mathbb{Z}$.
The set $\{0,3\}$ is a subgroup of $\mathbb{Z}_6$ under addition mod $6$.
Subgroup tests are important for understanding generated subgroups and cyclic groups.
Many subgroup proofs are easiest when written using algebraic expressions rather than checking elements one by one.