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PLease keep this to an intro abstract algebra level.

My book sayas that the set of all nonnegative integers (including 0) under addition is not a group because there is an identity element 0, but no inverse for 2

I don't understand what that part means.

Is it saying $2 + 0 \neq 0 + 2$?

Also for the natural numbers, why is there no identity element?

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2 Answers 2

up vote 3 down vote accepted

it is saying there is no nonnegative integer to add to 2 that gives 0

for the natural numbers there is no identity element because they are using a definition of natural numbers that doesn't include zero

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Why is 0 the identity element? My book claims that ex = xe = x, where e = 0 –  Hawk Sep 25 '12 at 6:25
    
@jak The * symbol indicates whatever operation is being considered. Since we are talking about the set of all nonnegative integers under addition, then * denotes addition in this context. Now do you see why 0 is the identity element? –  Ted Sep 25 '12 at 6:41

$0$ is an identity element because $n+0=0+n=n$ holds for all $n\in\mathbb N_0$. On the other hand, if $2$ had an inverse $x$, then we would have $2+x=x+2=0$. But there is (provably) no $x\in \mathbb N_0$ with this property.

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