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Section 2.1 Pre-read

A group is a pair \((G, \star)\text{,}\) where \(G\) is a collection of objects and \(\star\) is a binary operation on \(G\text{,}\) that satisfies the following four axioms. (Our textbook only lists three axioms, but the author is hiding an axiom that it is important to emphasize for beginning algebra students.)

Before we list the axioms, let's make sure we understand binary operation: a function that accepts two inputs and returns one output, all of the same type. So binary operation on \(G\) means that \(\star\) is a two-variable function whose two inputs and one output must both be objects from \(G\text{.}\) But instead of function notation \(\star(g_1,g_2)\) (where \(g_1,g_2\) are objects from \(G\)) we use “infix” notation: if \(g_1,g_2\) are objects from \(G\text{,}\) then we write \(g_1 \star g_2\) to mean the single output object produced by allowing \(\star\) to operate on the input pair \(g_1, g_2\text{.}\)

Here are the four axioms.

Group axioms.

  1. Closure.

    The collection \(G\) is closed under the operation \(\star\text{.}\) This means that when \(g_1,g_2\) is a pair of objects from \(G\text{,}\) then the operation result \(g_1 \star g_2\) is always some object from \(G\text{.}\)

  2. Associativity.

    The operation is associative. This means that when \(g_1,g_2,g_3\) is a triple of objects from \(G\text{,}\) then the results of computing \((g_1 \star g_2) \star g_3\) and \(g_1 \star (g_2 \star g_3)\) are always the same.

  3. Identity.

    The collection \(G\) contains an identity/unity/neutral element, denoted \(e\text{,}\) so that

    \begin{align*} e \star g \amp = g, \amp g \star e = g \end{align*}

    are both true for all objects \(g\) in \(G\text{.}\)

  4. Opposites.

    Each object \(g\) in the collection \(G\) has an corresponding opposite object \(\tilde{g}\) so that both

    \begin{align*} g \star \tilde{g} \amp = e, \amp \tilde{g} \star g = e \end{align*}

    are true.

The three most common situations in examples are outlined in Figure 2.1.1.

objects \(\star\) operation identity opposite
algebraic objects multiplication unit/identity/“one” object inverse
algebraic objects addition null/zero object negative
functions composition identity function \(\iota(x) = x\) inverse function
Figure 2.1.1. Patterns of example group operations.

Here are the axioms again in multiplicative notation.

Group axioms: multiplicative version.

  1. Closure.

    The collection \(G\) is closed under multiplication. That is, the resulting product \(g_1 g_2\) is always an object of \(G\) whenever the factors \(g_1,g_2\) are objects from \(G\text{.}\)

  2. Associativity.

    Multiplication is associative. This means that the results of computing \((g_1 g_2) g_3\) and \(g_1 (g_2 g_3)\) are always the same whenever \(g_1,g_2,g_3\) are objects from \(G\text{.}\)

  3. Identity.

    The collection \(G\) contains an identity/unity/neutral element, denoted \(e\) (but often denoted \(1\) or sometimes \(I\) in the multiplicative context) so that

    \begin{align*} e g \amp = g, \amp g e = g \end{align*}
    are both true for all objects \(g\) in \(G\text{.}\)

  4. Inverses.

    Each object \(g\) in the collection \(G\) has an corresponding inverse object \(\inv{g}\) so that both

    \begin{align*} g \inv{g} \amp = e, \amp \inv{g} g = e \end{align*}
    are true.

Warning 2.1.2.

In the context of a multiplicative group, an exponent of negative one does not mean reciprocal! That is, in general you should not rewrite \(\inv{g}\) as \(1 \over g\) when working with a multiplicative group. The notation \(\inv{g}\) is just a symbolic representation of the phrase “the inverse of the object \(g\) relative to the binary (multiplication) operation on \(G\text{.}\)”

In some contexts, it may acceptable to write inverses as fractions — for example, when working with a group of numbers. However, in many other contexts, fractions are meaningless — for example, when working with a group of matrices. Since fractions are not universally meaningful for multiplicative groups, it's best to never use fraction notation when working with an abstract multiplicative group.

And here are the axioms again in additive notation.

Group axioms: additive version.

  1. Closure.

    The collection \(G\) is closed under addition. That is, the resulting sum \(g_1 + g_2\) is always an object of \(G\) whenever the terms \(g_1,g_2\) are objects from \(G\text{.}\)

  2. Associativity.

    Addition is associative. This means that the results of computing \((g_1 + g_2) + g_3\) and \(g_1 + (g_2 + g_3)\) are always the same whenever \(g_1,g_2,g_3\) are objects from \(G\text{.}\)

  3. Identity.

    The collection \(G\) contains an identity/unity/neutral element, sometimes denoted \(e\) but usually denoted \(0\) in the additive context so that

    \begin{align*} 0 + g \amp = g, \amp g + 0 = g \end{align*}
    are both true for all objects \(g\) in \(G\text{.}\)

  4. Negatives.

    Each object \(g\) in the collection \(G\) has an corresponding negative object \(- g\) so that both

    \begin{align*} g + (- g) \amp = 0, \amp (- g) + g = 0 \end{align*}
    are true.

In the context of composition of functions, we will usually use multiplicative notation, so that we simply write \(f_1 f_2\) to mean \(f_1 \circ f_2\text{.}\) In this context, there always exists an identity function \(\iota(x) = x\text{,}\) but our collection \(G\) must actually contain this function to satisfy the third axiom. And for the fourth axiom, each function \(f\) must be one-to-one in order to have an inverse function \(\inv{f}\) (but again the collection \(G\) must also always contain this inverse in order to satisfy the fourth axiom).