In this statement and example

The set of all nonnegative integers (including 0) under addition is not a group. There is an identity element 0, but no inverse for 2

This is my confusion. Isn't 0, under addition, an inverse for all nonnegative integers? That is

a * a' = a'*a = a + a' = a' + a

Let $a = 2$ and $a' = 0$ and then $2 + 0 = 0 + 2$

Which is also doing the identity element (does this have to be unique for a group G to be a group?)

  • $\begingroup$ Yeah thank you. I overlooked that part $\endgroup$ – Hawk Oct 6 '12 at 21:50
  • $\begingroup$ This is kind of silly but when I first learned these, I viewed the inverse of an element as what I used to "kill" it, and the identity is what it became when it was "dead." Group elements, like people, are all destined for the same place - the grave - but we all get there by different methods. That's why it's the identity, but every element has its own, unique inverse. $\endgroup$ – Alexander Gruber Oct 6 '12 at 22:01

An inverse $b$ of $a$ is such that $ab = ba = e$ where $e$ is the identity element of the group. Can you find a positive integer $b$ such that $2+b = b+2 = 0$?

In the comment OP asked about the uniqueness of inverses. Yes, they must be unique: suppose there is another inverse $b'$ of $a$, and try to get to a contradiction.

  • $\begingroup$ Why did you set the operation to 0? $\endgroup$ – Hawk Oct 6 '12 at 21:47
  • $\begingroup$ Because 0 is the identity element. $\endgroup$ – Mike Oct 6 '12 at 21:49
  • $\begingroup$ @jak, as I say in the first sentence, the definition of the inverse is this: "$b$ is an inverse for $a$ (both elements of a monoid) iff $ab = ba = e$, where $e$ is the identity element of the monoid". $\endgroup$ – Andy Oct 6 '12 at 21:49
  • $\begingroup$ Oh okay, I overlooked that part. Throwing in something extra, does the inverse have to be unique? $\endgroup$ – Hawk Oct 6 '12 at 21:50

No, 0 isn't an inverse for all integers; it's the identity . Recall that the inverse $a'$ of element $a\in G$ has the property that $a+a' = a'+a=$ identity, which is in this case 0. For $a=2$, the inverse would be $-2$. Since $-2 \notin G$, $G$ isn't a group under addition.

The property you showed : $2+0=0+2=2$, illustrates that 0 is the identity element, not the inverse of 2. Hope this helps.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.