Understanding Identity Elements in Groups

Let’s consider a situation where two distinct identity elements, e and f, exist within a group G. We aim to demonstrate a contradiction, ultimately showing that e must equal f. For further context on group axioms, please refer to the previous article.

The foundational axioms of a group are as follows:

1. Closure property
2. Associativity
3. Identity element
4. Inverse element

Let g represent an element in G. Since e is recognized as an identity element, we have:

e * g = g

Similarly, since f is also an identity element, it follows that:

f * g = g

Given that both equations hold true, we can equate them:

e * g = f * g

According to the fourth axiom of group theory, each element must possess an inverse. Let h denote the inverse of g, and we can apply h to both sides of the equation e * g = f * g:

(e * g) * h = (f * g) * h

Utilizing the property of associativity (axiom 2), we can rearrange the parentheses without altering the outcome:

Thus, we can reformulate the equation:

e * (g * h) = f * (g * h)

Since h is the inverse of g, the expression g * h simplifies to the identity element:

e * (g * h) = e

The same principle applies to the right side of the equation:

f * (g * h) = f

By substituting these results into our earlier equation, we arrive at:

e = f

This conclusion reveals that e and f are not distinct identity elements; they are, in fact, the same element.

Conclusion

In summary, we have shown that within a group, there cannot be two distinct identity elements. The proofs we examined confirm the singular nature of the identity element, reinforcing the foundational principles of group theory.