Related Articles

# Elementary Properties of Groups

• Last Updated : 25 Feb, 2021

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.

Attention reader! Don’t stop learning now.  Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in GATE Test Series Course.

Learn all GATE CS concepts with Free Live Classes on our youtube channel.

My Personal Notes arrow_drop_up