# Two-Step Subgroup Test

## Contents

## Theorem

Let $\struct {G, \circ}$ be a group.

Let $H$ be a subset of $G$.

Then $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$ if and only if:

- $(1): \quad H \ne \O$, that is, $H$ is non-empty
- $(2): \quad a, b \in H \implies a \circ b \in H$
- $(3): \quad a \in H \implies a^{-1} \in H$.

That is, $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$ if and only if $\struct {H, \circ}$ is a $H$ be a nonempty subset of $G$ which is:

and:

## Proof

### Necessary Condition

Let $H$ be a subset of $G$ that fulfils the conditions given.

It is noted that the fact that $H$ is nonempty is one of the conditions.

It is also noted that the group product of $\struct {H, \circ}$ is the same as that for $\struct {G, \circ}$, that is, $\circ$.

So it remains to show that $\struct {H, \circ}$ is a group.

We check the four group axioms:

#### G0: Closure

The closure condition is given by condition $(2)$.

$\Box$

#### G1: Associativity

From Closed Substructure of Semigroup is Semigroup, associativity is inherited by $\struct {H, \circ}$ from $\struct {G, \circ}$.

$\Box$

#### G2: Identity

Let $e$ be the identity of $\struct {G, \circ}$.

From condition $(1)$, $H$ is non-empty.

Therefore $\exists x \in H$.

From condition $(3)$, $\struct {H, \circ}$ is closed under taking inverses

Therefore $x^{-1} \in H$.

Since $\struct {H, \circ}$ is closed under $\circ$, $x \circ x^{-1} = e = x^{-1} \circ x \in H$.

$\Box$

#### G3: Inverses

From condition $(3)$, every element of $H$ has an inverse.

$\Box$

So $\struct {H, \circ}$ satisfies all the group axioms, and is therefore a group.

So by definition $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$.

$\Box$

### Sufficient Condition

Now suppose $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$.

- $(1): \quad H \le G \implies H \ne \O$ from the fact that $H$ is a group and therefore can not be empty.
- $(2): \quad a, b \in H \implies a \circ b \in H$ follows from group axiom G0 (Closure) as applied to the group $\struct {H, \circ}$.
- $(3): \quad a \in H \implies a^{-1} \in H$ follows from group axiom G3 (Inverses) as applied to the group $\struct {H, \circ}$.

$\blacksquare$

## Also defined as

Some sources specify condition $(1)$ as being $e \in H$ in order to satisfy the non-emptiness condition, but from Identity of Subgroup it can be seen that this is not necessary, as this follows automatically by conditions $(2)$ and $(3)$.

Some sources completely omit to state the fact that $H$ needs to be non-empty.

Some sources, for example 1965: J.A. Green: *Sets and Groups*, use this property of subgroups as the definition of a subgroup, and from it deduce that a subgroup is a subset which is a group.

However, this definition is valid only if $\struct {G, \circ}$ is itself a group, as it is here so defined; it cannot be used to define a subgroup of a more general algebraic structure.

## Comment

This is called the **two-step subgroup test** although, on the face of it, there are three steps to the test.

This is because the fact that $H$ must be non-empty is usually an unspoken assumption, and is not specifically included as one of the tests to be made.

## Also see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 5.2$. Subgroups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 8$: Theorem $8.4: \ 1^\circ - 2^\circ$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.9$: Subgroups: Lemma $6$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): $1$: Subgroups: $1.\text{T}.6$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{II}$: Subgroups - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 36.3$: Subgroups - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Proposition $4.2$