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Answers to questions in additional notes on Group Theory

The numbers {1, -1} and addition.

We can see from the Cayley table that closure is not satisfied, nor is there an identity element. This is NOT a group.

The numbers {1, -1} and multiplication.

	1	-1
1	1	-1
-1	-1	1

Closure holds, the identity is 1, each element has an inverse (they are both self-inverse), and multiplication is associative. So this IS a group.

The symmetries of the equilateral triangle (there are six), and composition of symmetries.

These are the six symmetries:

Symbol	Operation	Symbol	Operation
e		p
a		q
b		r

We can use a Cayley table to check the axioms. As this is composition of symmetries, it matters which one we do first (unlike adding and multiplying numbers).

The Cayley table shows us that this is closed, that there is an identity (e), that each element as an inverse (e and the three reflections, p, q and r, are self-inverse, a and b are inverse to each other). Composition of symmetries of a plane figure is associative. So this IS a group.

The odd and even numbers, and addition.

We can see from the Cayley table that we have closure, an identity (even numbers), inverses (each is self-inverse), and we know that addition as associative, so this IS a group.

The odd and even numbers, and multiplication.

	even	odd
even	even	even
odd	even	odd

We can see from the Cayley table that we have closure, but no identity, and hence no inverses so this is NOT a group.

The real numbers without zero, and multiplication

If we multiply two real numbers together, the result must be a real number: there is no way you can multiply two real numbers together and produce something else, so this set with this operation is closed. The identity for multiplication is 1, and that is a real number. Each real number has an inverse under multiplication in the reals, eg. the inverse of 3 is 1/3 because ; the inverse of -0.75 is -4/3 because ; the inverse of is because is 1. If we take any element of the reals (without zero), say a, then 1/a is its inverse (why do we have to remove zero from our set of numbers?). All multiplication of numbers is associative. So this IS a group.

The integers and addition

If we add two integers together, the result must be another integer: there is no way you can add two integers together and produce something else, so this set with this operation is closed. The identity for addition is 0, and that is an integer. Each integer has an inverse under addition in the integers, eg. the inverse of 3 is -3 because ; the inverse of -199 is 199 because . If we take any element of the integers, say a, then -a is its inverse. All addition of numbers is associative. So this IS a group.

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