proof that e.g. integers with addition is a group How can we prove that a structure, such as integers with addition, is a group? I'm using group as an example of proving that a structure meets certain axioms, because the group axioms are relatively simple. I'm interested in general in how to prove that a particular structure meets a set of axioms.
I know how to prove for example for the integers with multiplication, that it is not a group (by showing that 0 has no inverse and therefore does not satisfy the 3rd group axiom), but proving that a structure is a group seems more difficult than proving that it is not a group.
 A: To show that the integers, $\mathbb{Z}$ together with usual addition form a group, you just need to check that the 4 properties* (or axioms, if you like) of a group are satisfied: 


*

*There exists an identity element in your group that fixes every element under the given binary operation.


Yes, the number $0$ is the identity since $0+k=k+0=k$,  $\forall k \in{\mathbb{Z}}$


*Closure: given any two elements $a$ and $b$ in the set, we need to show that $a+b$ stays in the set. 


Yes, given any two integers $a$ and $b$, their sum $a+b$ is again an integer.


*Associativity of the given operation.


Yes, addition in $\mathbb{Z}$ is associative i.e. $(a+b)+c=a+(b+c)$


*Existence of inverses: Given any element $a$ in the set, we need to find another element (call it "$a$ inverse") such that $a$ and its inverse commute and their operation together give the identity element.


Yes, given $a\in{\mathbb{Z}}$, then $-a$ is the inverse since $a+(-a) = (-a)+a=0$.
Since $\mathbb{Z}$ with addition satisfies 1-4 above, it is a group. Done.  
*These 4 can be economized into just 3 properties, depending on which book you may be using.
In general, if you want to prove that some other structure $\textit{is}$ what you think it should be (e.g. a ring, a topological space, a $\sigma$-algebra etc.) then the usual method of proof is to simply show that it satisfies each of the defining properties. Just work through and check off a list, just as I've done above.
Hope this helps.
