Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section11.3Some General Properties of Groups

In this section, we will present some of the most basic theorems of group theory. Keep in mind that each of these theorems tells us something about every group. We will illustrate this point with concrete examples at the close of the section.

One difficulty that students often encounter is how to get started in proving a theorem like this. The difficulty is certainly not in the theorem's complexity. It's too terse! Before actually starting the proof, we rephrase the theorem so that the implication it states is clear.

Proof

Next we justify the phrase “... the inverse of an element of a group.”

The same problem is encountered here as in the previous theorem. We will leave it to the reader to rephrase this theorem. The proof is also left to the reader to write out in detail. Here is a hint: If \(b\) and \(c\) are both inverses of \(a\text{,}\) then you can prove that \(b = c\). If you have difficulty with this proof, note that we have already proven it in a concrete setting in Chapter 5.

As mentioned above, the significance of Theorem 11.3.3 is that we can refer to the inverse of an element without ambiguity. The notation for the inverse of \(a\) is usually \(a^{-1}\) (note the exception below).

Example11.3.4Some Inverses

  1. In any group, \(e^{-1}\) is the inverse of the identity \(e\text{,}\) which always is \(e\text{.}\)

  2. \(\left(a^{-1}\right)^{-1}\) is the inverse of \(a^{-1}\) , which is always equal to \(a\) (see Theorem 11.3.5 below).

  3. \((x*y*z)^{-1}\) is the inverse of \(x * y * z\).

  4. In a concrete group with an operation that is based on addition, the inverse of \(a\) is usually written \(-a\). For example, the inverse of \(k - 3\) in the group \([\mathbb{Z}; +]\) is written \(-(k- 3)=3-k\). In the group of \(2 \times 2 \) matrices over the real numbers under matrix addition, the inverse of \(\left( \begin{array}{cc} 4 & 1 \\ 1 & -3 \\ \end{array} \right)\) is written \(-\left( \begin{array}{cc} 4 & 1 \\ 1 & -3 \\ \end{array} \right)\), which equals \(\left( \begin{array}{cc} -4 & -1 \\ -1 & 3 \\ \end{array} \right)\).

Again, we rephrase the theorem to make it clear how to proceed.

Proof

The next theorem gives us a formula for the inverse of \(a * b\text{.}\) This formula should be familiar. In Chapter 5 we saw that if \(A\) and \(B\) are invertible matrices, then \((A B)^{-1}= B^{-1} A^{-1}\).

Proof
Proof
Proof
Note11.3.10

Our proof of Theorem 11.3.9 was analogous to solving the concrete equation \(4x = 9\) in the following way: \[4 x=9=\left(4\cdot \frac{1}{4}\right)9=4\left(\frac{1}{4}9\right)\] Therefore, by cancelling 4, \[x = \frac{1}{4}\cdot 9 = \frac{9}{4}\]

If \(a\) is an element of a group \(G\text{,}\) then we establish the notation that \[a * a = a^2\quad \quad a*a*a=a^3\quad \quad \textrm{etc.}\] In addition, we allow negative exponent and define, for example, \[a^{-2}= \left(a^2\right)^{-1}\] Although this should be clear, proving exponentiation properties requires a more precise recursive definition.

Definition11.3.11Exponentiation in Groups

For \(n \geq 0\), define \(a^n\) recursively by \(a ^0 = e\) and if \(n > 0, a^n= a^{n-1} *a\). Also, if \(n >1\), \(a^{-n}= \left(a^n\right)^{-1}\).

Example11.3.12Some concrete exponentiations

  1. In the group of positive real numbers with multiplication, \[5^3= 5^2\cdot 5 =\left(5^1\cdot 5\right)\cdot 5=\left(\left(5^0\cdot 5\right)\cdot 5\right)\cdot 5=((1\cdot 5)\cdot 5)\cdot 5= 5 \cdot 5\cdot 5=125\] and \[5^{-3}= (125)^{-1}= \frac{1}{125}\]

  2. In a group with addition, we use a different form of notation, reflecting the fact that in addition repeated terms are multiples, not powers. For example, in \([\mathbb{Z}; +]\), \(a + a\) is written as \(2a\), \(a + a + a\) is written as \(3a\), etc. The inverse of a multiple of a such as \(- (a + a + a + a + a) = -(5a)\) is written as \((-5)a\).

Although we define, for example, \(a^5=a^4* a\), we need to be able to extract the single factor on the left. The following lemma justifies doing precisely that.

Based on the definitions for exponentiation above, there are several properties that can be proven. They are all identical to the exponentiation properties from elementary algebra.

Proof

Our final theorem is the only one that contains a hypothesis about the group in question. The theorem only applies to finite groups.

Proof

Consider the concrete group \([\mathbb{Z}; +]\text{.}\) All of the theorems that we have stated in this section except for the last one say something about \(\mathbb{Z}\text{.}\) Among the facts that we conclude from the theorems about \(\mathbb{Z}\) are:

  • Since the inverse of 5 is \(-5\), the inverse of \(-5\) is 5.

  • The inverse of \(-6 + 71\) is \(-(71) + -(-6) = -71 + 6\).

  • The solution of \(12 + x = 22\) is \(x = -12 + 22\).

  • \(-4(6) + 2(6) = (-4 + 2)(6) = -2(6) = -(2)(6)\).

  • \(7(4(3)) = (7\cdot 4)(3) = 28(3)\) (twenty-eight 3s).

Subsection11.3.1Exercises for Section 11.3

1

Let \([G; * ]\) be a group and \(a\) be an element of \(G\text{.}\) Define \(f:G \to G\) by \(f(x) = a * x\).

  1. Prove that \(f\) is a bijection.

  2. On the basis of part a, describe a set of bijections on the set of integers.

Answer
3

Prove by induction on \(n\) that if \(a_1\), \(a_2\), $\ldots $, \(a_n\) are elements of a group \(G\text{,}\) \(n\geq 2\), then \(\left(a_1*a_2*\cdots *a_n\right)^{-1}= a_n^{-1}*\cdots *a_2^{-1}*a_1^{-1}\). Interpret this result in terms of \([\mathbb{Z}; +]\) and \([\mathbb{R}^*;\cdot]\).

Answer
4

True or false? If \(a\text{,}\) \(b\text{,}\) \(c\) are elements of a group \(G\text{,}\) and \(a * b = c * a\), then \(b = c\). Explain your answer.

5

Prove Theorem 11.3.14.

Answer
6

Each of the following facts can be derived by identifying a certain group and then applying one of the theorems of this section to it. For each fact, list the group and the theorem that are used.

  1. \(\left(\frac{1}{3}\right)5\) is the only solution of \(3x = 5\).

  2. \(-(-(-18)) = -18\).

  3. If \(A, B, C\) are \(3\times 3\) matrices over the real numbers, with \(A + B = A + C\), then \(B = C\).

  4. There is only one subset of the natural numbers for which \(K \oplus A = A\) for every \(A \subseteq N\).