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Section 2.3 Subgroups and cosets

Subsection 2.3.1

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

\begin{equation*} gH := \{gh \colon h\in H\}. \end{equation*}

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

\begin{equation*} G/H := \{gH\colon g\in G\} \end{equation*}

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

\begin{equation*} D_4=\{R_0,R_{1/4},R_{1/2},R_{3/4},F_H,F_V,F_D,F_{D'}\} \end{equation*}
  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
  1. Yes. The composition of any two rotations is a rotation, and every rotation has an inverse that is also a rotation.
  2. 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.

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

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

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 \(G=\langle g\rangle\text{.}\)

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

  1. Find \(\langle F_H,F_V\rangle\subseteq D_4\text{.}\)
  2. Find \(\langle 6,8\rangle \subseteq \Z\text{.}\)
Answer
  1. \(\displaystyle \langle F_H,F_V\rangle=\{R_0,R_{1/2},F_H,F_V\}\)
  2. \(\displaystyle \langle 6,8\rangle =\langle 2\rangle = 2\Z\)

Exercises 2.3.2 Exercises

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

3.

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

4. Subgroups of \(\Z\) and \(\Z_n\).
  1. Let \(H\) be a subgroup of \(\Z\text{.}\) Show that either \(H=\{0\}\) or \(H=\langle d\rangle\text{,}\) where \(d\) is the smallest positive element in \(H\text{.}\)
  2. Let \(H\) be a subgroup of \(\Z_n\text{.}\) Show that either \(H=\{0\}\) or \(H=\langle d\rangle\text{,}\) where \(d\) is the smallest positive element in \(H\text{.}\)
  3. Let \(n_1,n_2,\ldots,n_r\) be positive integers. Show that
    \begin{equation*} \langle n_1,n_2,\ldots,n_r \rangle = \langle \gcd(n_1,n_2,\ldots,n_r)\rangle \end{equation*}

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 \(sa+tb=1\text{.}\)

5. Centralizers, Center of a group.

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

\begin{equation*} C(a) = \{g\in G\colon ag=ga\}. \end{equation*}

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

\begin{equation*} Z(G) = \{g\in G\colon ag=ga \;\; \forall a\in G\}. \end{equation*}
  1. Show that the centralizer \(C(a)\) of any element \(a\) in a group \(G\) is a subgroup of \(G\text{.}\)
  2. Show that the center \(Z(G)\) of a group \(G\) is a subgroup of \(G\text{.}\)
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

\begin{equation*} \langle g \rangle = \{g^0,g^1,g^2,\ldots,g^{n-1}\}\text{.} \end{equation*}

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

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

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

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.