Algebraic definition

Transcription

The first time you hear the definition of "group" you may think to yourself, "wheuuuuuhhh...?" It's an abstract idea, but what do you expect from abstract algebra? Because groups are abstract, they generalize a lot of different things from arithmetic, algebra, geometry and more. This generality is what makes groups so powerful. The mathematics you learn in abstract algebra can be applied to many different subjects. To get things started, let's think back to arithmetic. When studying arithmetic, you learn how to add, subtract, multiply and divide different kinds of numbers: integers, fractions, real numbers, complex numbers... Then one day, probably when you least expected it, you learned that subtraction is addition in disguise! For example, 7−4 is the same thing as 7 plus −4. So instead of subtracting, you're really adding a negative number. Similarly, division is multiplication in disguise! 9 divided by 5 is the same as 9 times ⅕. Instead of dividing, you're really multiplying by a fraction. So in arithmetic there are really just two operations: addition and multiplication. For addition, opposites are negative numbers. And for multiplication, opposites are reciprocals. In abstract algebra, we use the word "inverse" instead of "opposite." If you combine a number with its inverse, you get a special number called an identity element. For addition, the identity is 0. If you add 5 and its inverse -5, you get 0. What makes 0 unique is if you add it to any number, that number doesn't change. It retains its identity. For multiplication, if you multiply a number by its inverse - its reciprocal - you get 1. What makes 1 special is if you multiply it by any number, that number does not change. So for both addition and multiplication we have numbers. We have inverses. We have identity elements. Ladies and gentlemen, I think we're ready for the textbook definition of a group. Are you ready? A group is a set of elements... (Notice I said "elements" instead of "numbers". "Element" is more abstract.) The group has one operation. Let's use an asterisk to be more abstract. If you combine two elements in the group, the result is also in the group -- the group is "closed" under the operation. Each element has an inverse... If you combine an element with its inverse, you get an element we call the identity element. It's common to use the letter e for the identity. Lastly, we have associativity, because without associativity you couldn't solve the simplest of equations... Ladies and gentlemen, I give you the group! Let's get serious for a moment. Seriously. Groups do not have to be commutative. There are plenty of non-commutative groups in this crazy world. If a group IS commutative, we call it ... a commutative group. Another common name is abelian group.