# Elementary Properties of Groups

Let the set G on which a binary operation o is defined from a group (G , o). G is a group if it satisfies the following 3 properties:

- Associativity
- Identity
- Inverse

**Properties of Groups :**

**Property-1**:

If a , b, c ∈ G then, is a o b = a o c ⇒ b = c

**Proof: –**

Given a o b = a o c, for every a, b, c ∈ G Operating on the left with a^{-1}, where a^{-1}∈ G we have a^{-1}o (a o b) = a^{-1}o (a o c) or (a^{-1}o a) o b = (a^{-1}o a) o c [using associative property] or e o b = e o c, [using inverse property] or b = c, [using identity property]

Note that a o b is also written as ab.

This is known as the left cancellation law.

**Property-2:**

For every a ∈ G , e o a = a = a o e, where e is the identity element. i.e. The left identity element is also the right identity element.

**Proof: –**

If a^{-1}be the left inverse of a, then a^{-1}o (a o e) = (a^{-1}o a) o e [using associative property] or a^{-1}o (a o e) = e o e [using inverse property] = e [using identity property] or a^{-1}o (a o e) = a^{-1}o a [using inverse property] i.e. a^{-1}o (a o e) = a^{-1}o a

Hence, a o e = a by **property-1** i.e. left cancellation law. thus we find that e is also the right identity element and so it is called only the identity element.

**Property-3: **

For every a ∈ G , a^{-1} o a = e = a o a^{-1} i.e. the left inverse of an element is also its right inverse.

**Proof: –**

a^{-1}o (a o a^{-1}) = (a^{-1}o a) o a^{-1}[using identity property] = e o a^{-1 }[using inverse property] = a^{-1}o e [by property 2] i.e. a^{-1}o (a o a^{-1})= a^{-1}o e Hence, a o a^{-1}= e, by left cancellation law.

Thus, we find that the left inverse a^{-1} of element a is also its right inverse and so a^{-1} is called only the inverse of a.

**Property-4:**

If a , b, c ∈ G then, is b o a = c o a ⇒ b = c ** **

**Proof: –**

Given a o b = a o c, for every a, b, c ∈ G Operating on the left with a^{-1}, where a^{-1}∈ G we have (b o a) o a^{-1 }= (c o a) o a^{-1}or b o (a^{-1}o a) = c o (a^{-1}o a) [using associative property] or b o e = c o e, [using inverse property] or b = c, [using identity property]

This is known as right cancellation law.

**Property-5: **

For every a , b ∈ G we have (a o b)^{-1} = b^{-1} o a^{-1} i.e. The inverse of the product (or the composite) of two elements a, b of group G is the product (or composite) of the inverses of the two elements taken in the reverse order.

**Proof: – **

Let a^{-1}and b^{-1}be the inverses of a and b. Now,(a o b) o (b^{-1}o a^{-1}) = a o (b o b^{-1}) o a^{-1 }[using associative property] = a o e o a^{-1 }[using inverse property] = a o a^{-1 }[using identity property] = e [using inverse property] (a o b) o (b^{-1}o a^{-1}) = e Similarly, (b^{-1}o a^{-1}) o ( a o b)= e

Therefore, by the definition of inverse b^{-1} o a^{-1 }is the inverse of a o b. i.e. (a o b)^{-1}=b^{-1} o a^{-1}

This is known as the reversal rule.