9.1 Subgroups

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Definition 9.1:

A subgroup of a group \(G\) is a subset \(H \subseteq G\) that is itself a group with the same operation as \(G\). We sometimes write \(H \leq G\).

Remark:

It is important that the group operation is the same. For example \(\mathbb{R}\setminus\{0\}\) is a subset of \(\mathbb{R}\), but we do not regard \((\mathbb{R}\setminus\{0\},\times)\) as a subgroup of \((\mathbb{R},+)\) since the group operations are different.

Example:

For any group \(G\), the trivial subgroup \(\{e\}\) and \(G\) itself are subgroups of \(G\).

Definition 9.2:

We call a subgroup not equal to \(\{e\}\) a non-trivial subgroup, and a subgroup not equal to \(G\) a proper subgroup of \(G\).

Example:

We have \((\mathbb{Q},+)\) is a proper subgroup of \((\mathbb{R},+)\). We have \((\mathbb{Z},+)\) is a proper subgroup of both \((\mathbb{Q},+)\) and \((\mathbb{R},+)\).

Example:

The group \((2\mathbb{Z},+)\) is a subgroup of \((\mathbb{Z},+)\).

If \(n\) is a positive integer, then the group \((n\mathbb{Z},+)\) is a subgroup of \((\mathbb{Z},+)\).

Example:

The group of rotations of a regular \(n\)-sided polygon (i.e., \(\{e,a,a^2,\dots,a^{n-1}\}\)) is a subgroup of the dihedral group \(D_{2n}\).

Here’s a simple description of what needs to be checked for a subset to be a subgroup.

Theorem 9.3:

Let \(G\) be a multiplicatively written group and let \(H\subseteq G\) be a subset of \(G\), Then \(H\) is a subgroup of \(G\) if and only if the following conditions are satisfied.

• (Closure) If \(x,y\in H\) then \(xy\in H\).

• (Identity) \(e\in H\).

• (Inverses) If \(x\in H\) then \(x^{-1}\in H\).

Proof.

We don’t need to check associativity, since \(x(yz)=(xy)z\) is true for all elements of \(G\), so is certainly true for all elements of \(H\). So the conditions imply \(H\) is a group with the same operation as \(G\).

If \(H\) is a group, then (Closure) must hold. By uniqueness of identity and inverses, the identity of \(H\) must be the same as that of \(G\), and the inverse of \(x\in H\) is the same in \(H\) as in \(G\), so (Identity) and (Inverses) must hold.

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Proposition 9.4:

If \(H,K\) are two subgroups of a group \(G\), then \(H\cap K\) is also a subgroup of \(G\).

Proof.

We check the three properties in the theorem.

• If \(x,y\in H\cap K\) then \(xy\in H\) by closure of \(H\), and \(xy\in K\) by closure of \(K\), and so \(xy\in H\cap K\).

• \(e\in H\) and \(e\in K\), so \(e\in H\cap K\).

• If \(x\in H\cap K\), then \(x^{-1}\in H\) since \(H\) has inverses, and \(x^{-1}\in K\) since \(K\) has inverses. So \(x^{-1}\in H\cap K\).

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