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.
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