define a group and show that the cancellation law holds in a group
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The Cancellation Law for Groups
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The Cancellation Law for Groups
The Cancellation Law for Groups
On the Basic Theorems Regarding Groups page we looked at a whole bunch of theorems. We will now look at another important property of groups in that the cancellation law always applies.
Theorem 1: Let (G,∗) be a group and let a,b,c∈G. If a∗b=a∗c or b∗a=c∗a then b=c.
Proof: Let a−1∈G denote the inverse of a under ∗. Suppose that a∗b=a∗c. Then:
(1)
a∗b=a∗c(a−1∗a)∗b=(a−1∗a)∗ce∗b=e∗cb=c
Similarly, suppose now that b∗a=c∗a. Then:
(2)
b∗a=c∗a(b∗a)∗a−1=(c∗a)∗a−1b∗(a∗a−1)=c∗(a∗a−1)b∗e=c∗eb=c■
It is very important to note that the cancellation law holds with regards to the operation ∗ for any group (G,∗). We will see that the cancellation law does not necessarily hold with respect to an operation on a set when we look at algebraic structures with two defined operations.
It is also important to note that if a∗b=c∗a or b∗a=a∗c then we cannot necessarily deduce that b=c because we would then require the additional property that ∗ is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).
Fold
Table of Contents
The Cancellation Law for Groups
The Cancellation Law for Groups
On the Basic Theorems Regarding Groups page we looked at a whole bunch of theorems. We will now look at another important property of groups in that the cancellation law always applies.
Theorem 1: Let (G,∗) be a group and let a,b,c∈G. If a∗b=a∗c or b∗a=c∗a then b=c.
Proof: Let a−1∈G denote the inverse of a under ∗. Suppose that a∗b=a∗c. Then:
(1)
a∗b=a∗c(a−1∗a)∗b=(a−1∗a)∗ce∗b=e∗cb=c
Similarly, suppose now that b∗a=c∗a. Then:
(2)
b∗a=c∗a(b∗a)∗a−1=(c∗a)∗a−1b∗(a∗a−1)=c∗(a∗a−1)b∗e=c∗eb=c■
It is very important to note that the cancellation law holds with regards to the operation ∗ for any group (G,∗). We will see that the cancellation law does not necessarily hold with respect to an operation on a set when we look at algebraic structures with two defined operations.
It is also important to note that if a∗b=c∗a or b∗a=a∗c then we cannot necessarily deduce that b=c because we would then require the additional property that ∗ is commutative which is not one of the group axioms (but instead one of the Abelian group axioms).
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Answer :
Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :
- G is closed under *
- G is associative under *
- G has a unique identity element
- Every element of G has a unique inverse in G
Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .
To prove :
If a , b , c are any three arbitrary elements of a group G , then
ab = ac → b = c (left cancellation law)
ba = ca → b = c (right cancellation law)
Proof :
Let a⁻¹ ∈ G be the inverse element of a ∈ G , then we have a⁻¹a = aa⁻¹ = e , where e is the identity element in G .
Now , ab = ac
→ a⁻¹(ab) = a⁻¹(ac)
→ (a⁻¹a)b = (a⁻¹a)c
→ eb = ec
→ b = c
Also , ba = ca
→ (ba)a⁻¹ = (ca)a⁻¹
→ b(aa⁻¹) = c(aa⁻¹)
→ be = ce
→ b = c
Hence proved .
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