2.3: Subgroups and Cosets

Definition 2.3.1. Subgroups and cosets.

A subset \(H\) of a group \(G\) is called a subgroup of \(G\) if \(H\) itself is a group under the group operation of \(G\) restricted to \(H\text{.}\) We write \(H\leq G\) to indicate that \(H\) is a subgroup of \(G\text{.}\) A (left) coset of a subgroup \(H\) of \(G\) is a set of the form

\[
gH := \{gh \colon h\in H\}.
\nonumber \]

The set of all cosets of \(H\) is denoted \(G/H\text{.}\)

\[
G/H := \{gH\colon g\in G\}
\nonumber \]

Checkpoint 2.3.2.

Consider \(D_4\) as described in Checkpoint 2.1.6.

\[
D_4=\{R_0,R_{1/4},R_{1/2},R_{3/4},F_H,F_V,F_D,F_{D'}\}
\nonumber \]

1. Is the subset \(\{R_0,R_{1/4},R_{1/2},R_{3/4}\}\) of rotations a subgroup of \(D_4\text{?}\) Why or why not?
2. Is the subset \(\{F_H,F_V,F_D,F_{D'}\}\) of reflections a subgroup of \(D_4\text{?}\) Why or why not?

Answer

Yes. The composition of any two rotations is a rotation, and every rotation has an inverse that is also a rotation. No. Just observe that \(F_H^2=R_0\) is not a reflection. The group operation on \(D_4\) does not restrict properly to the subset of reflections.

Checkpoint 2.3.3.

Find \(G/H\) for \(G=S_3\text{,}\) \(H=\{e,(12)\}\text{.}\)

Answer

\begin{align*}
G/H & =\{eH, (12)H, (13)H, (23)H, (123)H, (132)H\}\\
& =
\{\{e,(12)\},\{(12),e\},\{(13),(123)\},\{(23),(132)\},\{(13),(123)\},\{(132),(23)\}\\
& = \{H,\{(13),(123)\},\{(23),(132)\}
\end{align*}

Proposition 2.3.4. Subgroup tests.

Let \(H\) be a subset of a group \(G\text{.}\) The following are equivalent.

1. \(H\) is a subgroup of \(G\)
2. (2-step subgroup test) \(H\) is nonempty, \(ab\) is in \(H\) for every \(a,b\) in \(H\) (\(H\) is closed under the group operation), and \(a^{-1}\) is in \(H\) for every a in \(H\) (\(H\) is closed under group inversion)
3. (1-step subgroup test) \(H\) is nonempty and \(ab^{-1}\) is in \(H\) for every \(a,b\) in \(H\)

Proof.

See Exercise 2.3.2.1.

Proposition 2.3.5. Subgroup generated by a set of elements.

Let \(S\) be a nonempty subset of a group \(G\text{,}\) and let \(S^{-1}\)S−1 denote the set \(\)S−1={\(\)s−1:s∈S} of inverses of elements in \(S\text{.}\) We write \(\langle S\rangle\) to denote the set of all elements of \(G\) of the form

\[
s_1s_2\cdots s_k
\nonumber \]

where \(k\) is a positive integer and each si is in \(S\cup S^{-1}\) for \(1\leq i\leq k\text{.}\) The set \(\langle S\rangle\) is a subgroup of \(G\text{,}\) called the subgroup generated by the set \(S\), and the elements of \(S\) are called the generators of \(\langle S\rangle\text{.}\)

Comment on notational convention: If \(S=\{s_1,s_2,\ldots,s_k\}\) is finite, we write \(\langle s_1,s_2,\ldots,s_k\rangle\) for \(\langle S\rangle\text{,}\) instead of the more cumbersome \(\langle \{s_1,s_2,\ldots,s_k\}\rangle\text{.}\)

Observation 2.3.6.

If \(G\) is a cyclic group with generator \(g\text{,}\) then \(G=\langle g\rangle\text{.}\)

Checkpoint 2.3.7.

Show that \(\langle S\rangle\) is indeed a subgroup of \(G\text{.}\) How would this fail if \(S\) were empty?

Checkpoint 2.3.8.

1. Find \(\langle F_H,F_V\rangle\subseteq D_4\text{.}\)
2. Find \(\langle 6,8\rangle \subseteq \mathbb{Z}\text{.}\)

Answer

1. \(\displaystyle \langle F_H,F_V\rangle=\{R_0,R_{1/2},F_H,F_V\}\)
2. \(\displaystyle \langle 6,8\rangle =\langle 2\rangle = 2\mathbb{Z}\)

Proposition 2.3.9. Cosets as equivalence classes.

Let \(G\) be a group and let \(H\) be a subgroup of \(G\text{.}\) Let \(\sim_H\) be the relation on \(G\) defined by \(x\sim_H y\) if and only if \(x^{-1}y \in
H\text{.}\) The relation \(\sim_H\) is an equivalence relation on \(G\text{,}\) and the equivalence classes are the cosets of \(H\text{,}\) that is, we have \(G/\!\!\sim_H= G/H\text{.}\)

Proof.

See Exercise 2.3.2.7

Corollary 2.3.10. Cosets as a partition.

Let \(G\) be a group and let \(H\) be a subgroup of \(G\text{.}\) The set \(G/H\) of cosets of \(H\) form a partition of \(G\text{.}\)

Exercises

Exercise 1.

Prove Proposition 2.3.4.

Exercise 2.

Find all the subgroups of \(S_3\text{.}\)

Answer

In the "list of values" permutation notation of Checkpoint 2.1.2, the subgroups of \(S_3\) are \(\{[1,2,3]\}\text{,}\) \(\{[1,2,3],[2,1,3]\}\text{,}\) \(\{[1,2,3],[1,3,2]\}\text{,}\) \(\{[1,2,3],[3,2,1]\}\text{,}\) and \(S_3\text{.}\) In cycle notation, the subgroups of \(S_3\) (in the same order) are \(\{e\}\text{,}\) \(\{e,(12)\}\text{,}\) \(\{e,(23)\}\text{,}\) \(\{e,(13)\}\text{,}\) \(\{e,(123),(132)\}\text{,}\) \(S_3\text{.}\)

Exercise 3.

Find all the cosets of the subgroup \(\{R_0,R_{1/2}\}\) of \(D_4\text{.}\)

Exercise 4.

Subgroups of \(\mathbb{Z}\) and \(\mathbb{Z}_n\).

1. Let \(H\) be a subgroup of \(\mathbb{Z}\). Show that either \(\)H={0} or \(\),H=⟨d⟩, where \(\)d is the smallest positive element in \(\).H.
2. Let \(H\) be a subgroup of \(\mathbb{Z}\). Show that either \(\)H={0} or \(\),H=⟨d⟩, where \(\)d is the smallest positive element in \(\).H.
3. Let \(n_1,n_2,\ldots,n_r\) be positive integers. Show that

   \[
   \langle n_1,n_2,\ldots,n_r
   \rangle = \langle \gcd(n_1,n_2,\ldots,n_r)\rangle
   \nonumber \]

Consequence of this exercise: The greatest common divisor \(\gcd(a,b)\) of integers \(a,b\) is the smallest positive integer of the form \(sa+tb\) over all integers \(s,t\text{.}\) Two integers \(a,b\) are relatively prime if and only if there exist integers \(s,t\) such that \(sa+tb=1\text{.}\)

Exercise 5.

Centralizers, Center of a group.

The centralizer of an element \(a\) in a group \(G\text{,}\) denoted \(C(a)\text{,}\) is the set

\[
C(a) = \{g\in G\colon ag=ga\}.
\nonumber \]

The center of a group \(G\text{,}\) denoted \(Z(G)\text{,}\) is the set

\[
Z(G) = \{g\in G\colon ag=ga \;\; \forall a\in G\}.
\nonumber \]

1. Show that the centralizer \(C(a)\) of any element \(a\) in a group \(G\) is a subgroup of \(G\)
2. Show that the center \(Z(G)\) of a group \(G\) is a subgroup of \(G\text{.}\)

Exercise 6.

The order of a group element.

Let \(g\) be an element of a group \(G\text{.}\) The order of \(g\text{,}\) denoted \(|g|\text{,}\) is the smallest positive integer \(n\) such that \(g^n=e\text{,}\) if such an integer exists. If there is no positive integer \(n\) such that \(g^n=e\text{,}\) then \(g\) is said to have infinite order. Show that, if the order of \(g\) is finite, say \(|g|=n\text{,}\) then

\[
\langle g \rangle =
\{g^0,g^1,g^2,\ldots,g^{n-1}\}\text{.}
\nonumber \]

Consequence of this exercise: If \(G\) is cyclic with generator \(g\text{,}\) then \(|G|=|g|\text{.}\)

Exercise 7.

Cosets of a subgroup partition the group, Lagrange's Theorem.

1. Prove Proposition 2.3.9.
2. Now suppose that a group \(G\) is finite. Show that all of the cosets of a subgroup H have the same size.
3. Prove the following.

Lagrange's Theorem.

If \(G\) is a finite group and \(H\) is a subgroup, then the order of \(H\) divides the order of \(G\text{.}\)

Hint

For part (b), let \(aH,bH\) be cosets. Show that the function \(aH\to bH\) given by \(x\to ba^{-1}x\) is a bijection.

Exercise 8.

Consequences of Lagrange's Theorem.

1. Show that the order of any element of a finite group divides the order of the group.
2. Let \(G\) be a finite group, and let \(g\in
   G\text{.}\) Show that \(g^{|G|}=e\text{.}\)
3. Show that a group of prime order is cyclic.