Groups

Section 1.1 Groups

This first quarter of Math 331 is devoted entirely to the study of groups. This a fitting place to begin both as groups represent a type of algebraic object with near minimal structure and are omnipresent in the world of mathematics and the empirical sciences more generally. More conceptually, groups are intimately connected with the notion of symmetry in mathematics. Put plainly, wherever you find mathematical discussion of symmetry, a group usually lurks in the background as the thing that defines that symmetry. We will see plenty of examples of this in the coming days. In the meantime, for an excellent elaboration of this theme, check out Keith Conrad’s blurb Why groups?

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Subsection 1.1.1 Definition

Definition 1.1.1. Groups.

A group is an ordered pair \((G,\cdot)\) where \(G\) is a nonempty set and \(\cdot\) is a binary operation

\begin{align*} G\times G \amp \xrightarrow{\phantom{XX}\bullet\phantom{XX}} G\\ (g,h) \amp \longmapsto g\cdot h\text{,} \end{align*}

satisfying the following axioms.

Associativity.

For all \(g_1,g_2,g_3\in G\) we have

\begin{equation} (g_1\cdot g_2)\cdot g_3 = g_1\cdot (g_2\cdot g_3)\text{.}\tag{1.1.1} \end{equation}

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Group identity.

There is an element \(e\in G\text{,}\) such that for all \(g\in G\text{,}\) we have

\begin{equation} g\cdot e = e\cdot g = g\text{.}\tag{1.1.2} \end{equation}

We call \(e\) the group identity of \(G\text{.}\)

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Group inverses.

For all \(g\in G\) there is an element \(g^{-1}\in G\) satisfying

\begin{equation} g\cdot g^{-1} = g^{-1}\cdot g = e\text{.}\tag{1.1.3} \end{equation}

We call \(g^{-1}\) the group inverse of \(g\text{.}\)

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Fiat 1.1.2. Group law notation.

You are free to denote the binary group law in any manner you like. Often, as in Specimen 1–2, we will make a notational choice that reflects the specific nature of a given group.

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Furthermore, with the exception of the case where \(+\) is used to denote the group law, we will often omit the group operation symbol when writing expressions involving group elements.

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Specimen 1. Ring additive groups.

Let \(R\in \{\Z,\Q, \R, \C\}\text{,}\) and let \(+\) be the usual addition operation on \(R\text{.}\) The pair \((R,+)\) is a group, called the additive group of \(R\text{.}\)

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The group identity of \(R\) is \(0\text{,}\) also known as the additive identity of \(R\text{.}\)

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Given any \(r\in R\text{,}\) its group inverse is its minus, \(-r\text{,}\) also known as the additive inverse \(r\text{.}\)

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Specimen 2. Ring multiplicative groups.

Let \(R\in \{\Z,\Q, \R, \C\}\text{.}\) An element \(r\in R\) is a unit (or (multiplicatively) invertible if there is an element \(r^{-1}\in R\) satisfying \(r\,r^{-1}=r^{-1}\, r=1\text{.}\) In this case \(r^{-1}\) is called the multiplicative inverse of \(r\text{.}\) Define \(R^*\) to be the set of all units of \(R\text{:}\) i.e.,

\begin{equation*} R^* = \{ r\in R \mid \text{there exists } r^{-1}\in R \text{ such that } r\, r^{-1}=r^{-1}\, r = 1 \}\text{.} \end{equation*}

If \(R\in \{\Q, \R, \C\}\text{,}\) then \(R^*=R-\{0\}\text{,}\) since in these number systems all nonzero elements have a multiplicative inverse. By contrast, we have

\begin{equation*} Z^*=\{1, -1\}\text{,} \end{equation*}

since \(1\) and \(-1\) are the only integers that have an integer multiplicative inverse.

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The \((R^*,\cdot)\text{,}\) where \(\cdot\) is the usual multiplciation operation on \(R\text{,}\) is a group called the multiplicative group of \(R\text{.}\)

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The group identity of \(R^*\) is \(1\text{,}\) also known as the multiplicative identity of \(R\text{.}\)

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Given any \(r\in R^*\text{,}\) its group inverse is the multiplicative inverse \(r^{-1}\) defined above.

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Definition 1.1.3. Abelian group.

A group \((G,\cdot)\) is abelian (or commutative) if

\begin{equation*} g\, h=h\, g \end{equation*}

for all \(g,h\in G\text{.}\) The group is nonabelian (or noncommutative), if it is not abelian.

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Specimen 3. General linear group.

Let \(R\in \{\Q, \R, \C\}\text{.}\) The set of all \(n\times n\) matrices with coefficients in \(R\) is denoted \(\ML_n(R)\text{.}\) We denote by \(\GL_n(R)\) the set of invertible \(n\times n\) matrices with coefficients in \(R\text{:}\) i.e.,

\begin{align*} \GL_n(R) \amp \end{align*}

is denoted \(\GL_n(R)\text{:}\) i.e.,

\begin{align*} \GL_n(R) \amp = \{A\in M_n(R)\mid A \text{ is invertible} \} \\ \amp =\{A\in M_n(R)\mid \det A\ne 0\}\text{.} \end{align*}

The pair \((\GL_n(R), \cdot)\text{,}\) where \(\cdot\) is the usual matrix multiplication operation, is a group called the general linear group of degree \(n\) over \(R\text{.}\)

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The group identity of \(\GL_n(R)\) is the \(n\times n\) identity matrix \(I\text{.}\)

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Given any \(A\in \GL_n(R)\text{,}\) its group inverse is the matrix inverse \(A^{-1}\text{.}\)

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Example 1.1.4. \(\GL_n(R)\) is nonabelian.

Let \(R\in \{\Q,\R, \C\}\) and let \(n\geq 2\text{.}\) Prove that \(\GL_n(R)\) is a nonabelian group.

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Solution.

Let

\begin{align*} A \amp = \begin{bmatrix} 1 \amp 1 \\ 0 \amp 1 \end{bmatrix} \amp B\amp= \\begin{bmatrix} 1 \amp 0 \\ 1 \amp 1 \end{bmatrix}\text{.} \end{align*}

We have \(A, B\in \GL_2(R)\) (easy determinant computation), and furthermore

\begin{align*} AB \amp = \begin{bmatrix} 2 \amp 1 \\ 1 \amp 1 \end{bmatrix} \\ BA \amp = \begin{bmatrix} 1 \amp 1 \\ 1 \amp 2 \end{bmatrix} \text{.} \end{align*}

Since \(AB\ne BA\text{,}\) we conclude that \(\GL_2(R)\) is nonabelian.

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Next, given any \(n\geq 3\text{,}\) consider the block matrices

\begin{align*} A' \amp = \begin{bmatrix} A\amp 0 \\ 0\amp I_{n-2} \end{bmatrix} \amp B'\amp=\begin{bmatrix} B\amp 0 \\ 0\amp I_{n-2} \end{bmatrix} \text{,} \end{align*}

where \(A\) and \(B\) are the \(2\times 2\) matrices above. From properties of block matrix arithmetic it follows that \(A'\) and \(B'\) are invertible (in fact, \(\det A'=\det B'=1\)) and

\begin{align*} A'B' \amp = \begin{bmatrix} AB\amp 0 \\ 0\amp I_{n-2} \end{bmatrix} \amp B'A'\amp=\begin{bmatrix} BA\amp 0 \\ 0\amp I_{n-2} \end{bmatrix} \text{.} \end{align*}

Since \(AB\ne BA\text{,}\) we conclude that \(\GL_n(R)\) is nonabelian, as desired.

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Subsection 1.1.2 Product groups

The group product operation is an easy and important manner of constructing new groups from a collection of existing ones.

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Definition 1.1.5. Product group.

Given groups \(G\) and \(H\text{,}\) the product group \(G\times H\) is the group with underlying set

\begin{equation*} G\times H=\{(g,h)\mid g\in G, h\in H\} \end{equation*}

and group operation

\begin{equation*} (g,h)\cdot (g',h')=(gg', hh')\text{.} \end{equation*}

More generally, given a family of groups \((G_i)_{i\in I}\) indexed by a nonempty set \(I\text{,}\) their product is the group \(\prod_{i\in I}G_i\) with underling set

\begin{equation*} \prod_{i\in I}G_i=\{(g_i)_{i\in I}\mid g_i\in G_i \text{ for all } i\in I\} \end{equation*}

and group operation

\begin{equation*} (g_i)_{i\in I}\cdot (h_i)_{i\in I}=(g_ih_i)_{i\in I} \end{equation*}

for all \(g=(g_i)_{i\in I}\) and \(h=(h_i)_{i\in I}\) in \(\prod_{i\in I}G_i\text{.}\)

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We write \(G^n\) for the \(n\)-fold group product

\begin{equation*} \underset{n \text{ times}}{\underbrace{G\times G\times \cdots \times G}} \end{equation*}

of a group \(G\) with itself.

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Remark 1.1.6. Arbitrary products.

See Tuples and Cartesian product for more details about tuples and Cartesian products indexed by arbitrary sets \(I\text{.}\)

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Proposition 1.1.7. Product group.

Given a family \((G_i)_{i\in I}\) of groups indexed by a nonempty set \(I\text{,}\) the product \(\prod_{i\in I} G_i\) is a group with respect to the binary operation defined in Definition 1.1.5.

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Proof.

We verify each group axiom in turn.

Given elements \(g=(g_i)_{i\in I}\text{,}\) \(h=(h_i)_{i\in I}\text{,}\) and \(k=(k_i)_{i\in I}\) of \(\prod_{i\in I} G_i\text{,}\) we have

\begin{align*} g(hk) \amp = (g_i)_{i\in I} \cdot \left( (h_i)_{i\in I}\cdot (k_i)_{i\in I}\right)\\ \amp = (g_i)_{i\in I} \cdot (h_i k_i)_{i\in I}\\ \amp = (g_i (h_ik_i))_{i\in I}\\ \amp = ((g_ih_i)k_i)_{i\in I} \amp (\text{assoc. of } G_i)\\ \amp = (g_ih_i)_{i\in I}\cdot (k_i)_{i\in I}\\ \amp = \left((g_i)_{i\in I}\cdot (h_i)_{i\in I}\right)\cdot (k_i)_{i\in I}\\ \amp =(gh)k\text{.} \end{align*}

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Let \(e_i\) denote the identity element of \(G_i\) for each \(i\in I\text{,}\) and let \(e=(e_i)_{i\in I}\text{.}\) For any \(g=(g_i)_{i\in I}\text{,}\) we have

\begin{align*} g\cdot e \amp = (g_i)_{i\in I} \cdot (e_i)_{i\in I}\\ \amp = (g_ie_i)_{i\in I}\\ \amp = (g_i)_{i\in I}\\ \amp = g \end{align*}

and

\begin{align*} e\cdot g \amp = (e_i)_{i\in I} \cdot (g_i)_{i\in I}\\ \amp = (e_ig_i)_{i\in I}\\ \amp = (g_i)_{i\in I}\\ \amp = g\text{,} \end{align*}

showing that \(e=(e_i)_{i\in I}\) satisfies the group identity axiom.

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Given \(g=(g_i)_{i\in I}\in \prod_{i\in I}G_i\text{,}\) let \(g^{-1}=(g_i^{-1})_{i\in I}\text{.}\) We have

\begin{align*} g\cdot g^{-1} \amp = (g_i)_{i\in I} \cdot (g_i^{-1})_{i\in I}\\ \amp = (g_ig_i^{-1})_{i\in I}\\ \amp = (e_i)_{i\in I}\\ \amp = e \end{align*}

and

\begin{align*} g^{-1}\cdot g \amp = (g_i^{-1})_{i\in I} \cdot (g_i)_{i\in I}\\ \amp = (g_i^{-1}g_i)_{i\in I}\\ \amp = (e_i)_{i\in I}\\ \amp = e\text{,} \end{align*}

showing that \(g^{-1}\) is a group inverse of \(g\text{.}\)

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Subsection 1.1.3 Elementary properties

Proposition 1.1.8. Group properties.

Let \((G,\cdot)\) be a group.

Cancellation laws.
Let \(g,h,k\in G\text{.}\)
1. \(gh=gk\) if and only if \(h=k\text{.}\)
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2. \(hg=kg\) if and only if \(h=k\text{.}\)
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Uniqueness of identity.
The identity element of \(G\) is unique: i.e., there is exactly one element \(e\in G\) satisfying (1.1.2).
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Uniqueness of inverse.
Every \(g\in G\) has a unique inverse element: i.e., for each \(g\in G\) there is exactly one element \(g^{-1}\in G\) satisfying (1.1.3).
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\((g^{-1})^{-1}=g\) for all \(g\in G\text{.}\)
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Inverse of product.
\((g\, h)^{-1}=h^{-1}\, g^{-1}\) for all \(g,h\in G\text{.}\)
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Generalized associativity.

Any two parenthetical groupings of an \(n\)-tuple \((g_1,g_2,\dots, g_n)\in G^n\) give rise to the same product in \(G\text{,}\) which we denote as

\begin{equation*} g_1g_2\cdots g_n\text{.} \end{equation*}

For example, we have

\begin{align*} ((g_1\, g_2)\, g_3)\, g_4 \amp = (g_1\, (g_2\, g_3))\, g_4 \\ \amp = g_1\, ((g_2\, g_3)\, g_4)\\ \amp = g_1\, (g_2\, (g_3\, g_4))\\ \amp =(g_1\, g_2)(g_3\, g_4) \end{align*}

for any \((g_1,g_2,g_3,g_4)\in G^4\text{,}\) and we denote this common element \(g_1\,g_2\,g_3\,g_4\text{.}\)

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Proof.

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Definition 1.1.9. Group exponentiation.

Let \(G\) be a group. Given \(n\in \Z\) and \(g\in G\text{,}\) the \(n\)-th power of \(g\text{,}\) denoted \(g^n\) is defined via cases as follows.

\(\displaystyle g^0=e\)
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Given positive integer \(n\text{,}\) we define

\begin{align*} g^n \amp =\underset{n \text{ times}}{\underbrace{g\, g\cdots g}}\amp g^{-n} \amp=\underset{n \text{ times}}{\underbrace{g^{-1}\, g^{-1}\cdots g^{-1}}}\text{.} \end{align*}

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Definition 1.1.10. Group element order.

Let \(g\) be an element of the group \(G\text{.}\) The order of \(g\text{,}\) denoted \(\ord g\) is defined as follows.

If there is no positive integer \(n\text{,}\) such that \(g^n=e\text{,}\) then \(g\) has infinite order, denoted \(\ord g=\infty\text{.}\)
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If there is a positive integer \(k\) such that \(g^k=e\text{,}\) then \(\ord g\) is defined as the smallest positive integer \(n\) satisfying \(g^n=e\text{.}\)
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Example 1.1.11. Order of inverse.

Let \(g\) be an element of the group \(G\text{.}\) Prove: \(\ord g=\ord g^{-1}\text{.}\)

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Example 1.1.12. Element of infinite order.

Assume \(g\) is an element of the group \(G\) of infinite order. Prove that the elements \(g^n\text{,}\) \(n\in \Z\text{,}\) are all distinct.

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