Binary operations and groups

Maths · Sets, functions and groups. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A binary operation * on a non-empty set G combines any two elements of G to give one element that is again in G. This is the closure property. Subtraction is not a binary operation on N, because 2 − 5 is not in N.

An operation can then be tested for other properties: commutative, associative, an identity element, and an inverse for each element.

The more properties that hold, the more special the name of the structure: groupoid, semi-group, monoid, group and abelian group. To test for a group, check the properties one by one in that order.

Rules to remember

Commutative: a * b = b * a. Associative: (a * b) * c = a * (b * c), for all elements of G.
Identity e: a * e = e * a = a for every a in G. Inverse a′ of a: a * a′ = a′ * a = e.
Groupoid: closed under *. Semi-group: closed and associative. Monoid: a semi-group with an identity.
Group: closed, associative, has an identity, and every element has an inverse in the set.
Abelian group: a group in which * is also commutative.
(Z, +) is a group with identity 0; N and W under + are not, because inverses are missing.
{1, −1, i, −i} is an abelian group under multiplication, with identity 1.

Common mistake

Students assume the identity is always 0 or 1. For a newly defined operation, find e from the equation a * e = a, and find an inverse from a * x = e with that e.

Worked example

On the set of real numbers a * b = a + b + ab. Find the identity element and the inverse of 3.

Identity: a * e = a gives a + e + ae = a, so e(1 + a) = 0 for every a, and e = 0.
Inverse of 3: 3 * x = 0 gives 3 + x + 3x = 0, so 4x = −3 and x = −3/4.
Check: 3 + (−3/4) + 3 × (−3/4) = 3 − 3/4 − 9/4 = 0.

Answer: Identity = 0; inverse of 3 = −3/4

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