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On my scholars book it says that:

Inverse should exist for a group formation. Let * be the binary operation of the group. Then there should be $a$ and $c$ in our group $G$ such that $a*c=0$

So something that I didn't understand is that why it should be equal to $0$?. In groups with multiplication with the binary operation we look for 1/a such that $a(1/a)=1$ and we say 1/a is an inverse when the multiplication equals to $1$. But from this definition I understand that the solution should be zero for an inverse to exist in ANY binary operation? Is the definition wrong? Or is the binary operation meant here is summation? But why show the operation as $*$?

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    $\begingroup$ The entire definition is rather far from correct anyway. What book is this? $\endgroup$ – Tobias Kildetoft Sep 29 '16 at 18:05
  • $\begingroup$ To repair the definition you quoted, one should say that $0$ represents the identity element of the group (not necessarily the number zero, so it would probably be better to use a different symbol instead of $0$), and one should replace "there should be $a$ and $c$ in our group" with "for every $a$ in our group there should be some $c$ in our group". $\endgroup$ – Andreas Blass Sep 29 '16 at 18:11
  • $\begingroup$ You should be careful not to carry over all of the ideas from what you know into group theory. The symbol $*$ is a general binary operation, it doesn't mean ``multiplication" in the same sense that it does in $\mathbb{Z}$ for instance, it's just an arbitrary operation. $\endgroup$ – walkar Sep 29 '16 at 18:12
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Zero is just a common notation for the identity in ableian groups; you may also see $$\text{Id}, \text{Id}_{G}, 1, 1_G, e,$$ etc. All refer to the identity element in a group $G$.

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" Then there should be $a$ and $c$ in our group G such that $a∗c=0$"

Ooh, that isn't quite worded right.

The group is an abstract set that could be anything.

One of the elements is what we call $0_g$ (I put the $_g$ on the zero so you don't mistake it for the "real" zero.

And $0_g$ has the property that for any $a$ in the group $a * 0_g = 0_g * a = a$. We call $0_g$ the "identity element".

Now to your statement.

" Then there should be $a$ and $c$ in our group G such that $a∗c=0$"

Not quite there aren't just one pair. The are lots of pairs.

For every element $a$ there exists an element which we will call $a^{-1}$ that will have the property $a*a^{-1} = a^{-1}*a = 0_g$. We call this "the inverse of $a$".

". In groups with multiplication with the binary operation we look for 1/a such that a(1/a)=1 and we say 1/a is an inverse when the multiplication equals to 1."

Yes! That is the perfect example. In this case $0_g = 1$. Remember, the group is abstract and can be anything the fits the rules.

"But from this definition I understand that the solution should be zero for an inverse to exist in ANY binary operation?"

Yep! "zero" in this case is the number 1.

Here's another example: let's suppose I live in a slightly surreal universe where you can mix any two foods together to get another food and there's this strange property that for every food there is a food that neutralizes it.

So chicken + turmeric = curry. But curry + milk = chicken. turmeric and milk cancel each other out. (So milkshake + turmeric = ice cream... okay, it's a pretty surreal universe... let's just go with it....)

Now suppose $<<$anything$>>$ + air = $<<$anything$>>$. Then $0_{\text{food}} = $ air.

And turmeric + milk = air = $0_g$.

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