A Mathematical Group ☆

While many future mathematics teachers may never step foot in an abstract algebra course, those that do are often initially presented with the axiomatic definition of a group (closure, associativity, identity element, inverse elements) and then presented with some common examples of group tables – frequently small finite sets – to illustrate the impact of the axioms. Unfortunately, doing so hides the essence of why these collective properties form a necessary structure for algebra and algebraic reasoning.

In order to communicate why these individual properties are important collectively, one possible activity that I have used is to elaborate on solving simple, single-step, equations. In class, it is common to use a “single step” to solve a simple equation, for example, x+5=12; however, there are actually four assumptions being made about how the operation addition works on the set of real numbers. These assumptions, collectively, are important for algebraic reasoning.

x + 5 = 12

(x + 5) + -5 = 12 + -5

x +(5 + -5) = 12 + -5 (Associativity (of addition on R))

x + 0 = 12 + -5 (Inverse elements (of addition on R))

x = 12 + -5 (Identity element (of addition on R))

x = 7 (Closure (of addition on R))

Without these assumptions, of associativity, inverse elements, identity element, and closure, the algebraic solving process may not generalize. While in this case we make use of -5, in the more general case, it would have to be true that every element had an inverse element. Similarly, while it is 0 under addition, without an identity element the solving process would loop infinitely: the identity element is the key to transforming an unknown sum (x+5) into a known sum (x+0). Also, while in this case the sum of 12 and -5 is a number, in the more general case, it would have to be true that the sum of any two elements produced another number. (We note that commutativity is not required to be a group, but is required for other important algebraic structures, such as a field.)

Indeed, perhaps the more powerful illustration of these four properties working collectively as a foundation for algebra is to solve an equation on an unfamiliar set and operation. For example, what is the solution to the equation X ° RX = R2? (Based on the operation table below, e.g., R1° RY = RZ.)

Group Table

While searching the table for the solution is a valid approach to find the value for X, this more abstract case can be used to present the collective impact of these four axioms of a group. First, I note that this table is analogous to the composition of the symmetries of a triangle, which indeed is a group. (Another option is to verify each of these four properties based only on the table above.) The operation is closed on this set (as the composition of any two elements forms an element in the set); the identity element is R0; each element has a unique inverse element that produces R0; and composition is associative. Therefore, to find X algebraically, it is necessary only to apply each of these properties in turn, and compute the result of one composition (R2 ° RX), to solve for X.

X ° RX = R2

(X ° RX) ° RX = R2 ° RX

X ° (RX ° RX) = R2 ° RX  (Associativity (of composition on the set of triangle symmetries))

X ° R0  = R2 ° RX  (Inverse Elements (of composition on the set of triangle symmetries))

X = R2 ° RX (Identity Element (of composition on the set of triangle symmetries))

X = RZ  (Closure (of composition on the set of triangle symmetries))

Indeed, as evident from the example above of solving a simple equation, it is not just the individual arithmetic properties – such as associativity, identity element, inverse elements, closure – that are meaningful in mathematics, but rather their collective importance as they become a necessary structure for algebra and algebraic reasoning. It is this basic structure of a group that turns the process of “guess and check” (which was the natural inclination for searching the table in the abstract example, as well as students’ tendency when first introduce to solving equations) into systematic reasoning, based on the collection of arithmetic properties.