Prove that the identity element e of a group G is unique.
An identity element can be a left-identity (1*x=x, x∈R) a right-identity (a^1=a, x∈R) or both (1*x=x, x*1=x, x∈R).
If the group (S,*) is such that it has a left (e1) and right (e2) identities, then it follows that
e1*x=x ∀x∈S, and
x=x*e2 ∀x∈S
From which we conclude:
e1=e1*e2=e2 meaning that e1=e2, or the (two-sided) identity element must be unique.
To prove that the identity element e of a group G is unique, we need to show that there cannot be two distinct elements in G that both satisfy the properties of the identity element.
Let's assume that G is a group with two distinct identity elements, denoted e1 and e2.
First, we will show that e1 = e1 * e2.
Since e2 is an identity element, for any element a in G, we have a * e2 = a.
Substituting a = e1 gives e1 * e2 = e1.
Since e1 is also an identity element, we have e1 * e2 = e2.
Combining the two equations, we get e1 = e2.
Next, we will show that e1 = e2 * e1.
Since e2 is an identity element, for any element a in G, we have a * e2 = a.
Substituting a = e1 gives e1 * e2 = e1.
Since e1 is also an identity element, we have e1 * e2 = e2.
Combining the two equations, we get e1 = e2.
Therefore, we have shown that e1 = e2 * e1 and e1 = e1 * e2.
By the cancellation law, we can conclude that e2 * e1 = e2 and e2 * e1 = e1.
Therefore, e2 = e1.
This contradicts our assumption that e1 and e2 are distinct identity elements.
Hence, we can conclude that the identity element e in a group G is unique.
To prove that the identity element, denoted as 'e', of a group G is unique, we need to show that there cannot be more than one element in G that satisfies the properties of an identity element.
To do this, let's assume that G has two identity elements, e1 and e2. We need to prove that e1 = e2.
By definition, an identity element is an element that, when combined with any other element in the group, leaves that element unchanged.
So, let's consider the element a in group G. We know that e1 is an identity element, so by the definition of identity, we have:
a * e1 = a
Now, since e2 is also an identity element, we can replace e1 with e2:
a * e2 = a
But, by the definition of identity, we also know that e2 * a = a. Therefore:
e2 * a = a
Now, we can apply the cancellation law of groups, which states that if b * x = b * y, then x = y. So, using this law, we can cancel 'a' from both sides of the equation:
e2 = e1
Hence, we have shown that if G has two identity elements e1 and e2, then e1 must be equal to e2. Therefore, the identity element of a group G is unique.
Note: In this proof, we relied on the properties of groups, such as closure, associativity, and inverses. These properties are essential for groups and are assumed to be true when working with them.