This question is in relation to Group Theory.
I am trying to determine which of the following pairs consisting of a set and a binary operation ($G$, *), is a group. And which are Abelian groups.
$1.$ $G=\mathbb R^3 , x.y:= x$ x $y$, the cross product.
$2.$ $G={x+\sqrt 2 y : x,y \mathbb Z}$ where $a*b : a+b$, the usual addition
$3.$ $G={x+\sqrt 2 y : x,y \mathbb Q}$ where $a*b : a+b$, the usual addition
$4.$ $G={x+\sqrt 2 y : x,y \mathbb Q}$ where $a*b : a+b$, the usual multiplication
$5.$ $G=({x+\sqrt 2 y : x,y \mathbb Q})$ \{$0$} where $a.b : a+b$, the usual multiplication
My understanding: I am new to this topic but I assume here i need to prove associativity here, that is that for any $3$ elements $a*(b*c)=(a*b)*c$
I would also need to show the existence of an identity element, that is $e*a=a*e=a$ where $e$ is the identity element.
And finally I would need to prove that there is closure under inverses, $a*a^{-1}=e$
In order to prove if these groups are abelian, I would need to show communtivity, where $a*b=b*a$
Is my understanding correct and if so how do i apply these axioms. A solution to one part would really help me to understand the work needed to be done to solve the rest.
HINT
The most awkward result to prove in questions of this type is always associativity. However, here all but one of the operations is "the usual multiplication/addition" of reals and so you know these will be associative. That leaves the vector product which is not associative, so just find an example of three vectors to show this. That will prove (1) is not a group and you then need do nothing further for that case.
Can you proceed now? If you show your work it will be easy to help you further.
P.S. Your understanding of the problem is correct except that you say you need to prove "closure under inverses" but what you actually must prove is the existence of inverses and also you must prove closure in general.
As an added help you should note that "the usual multiplication/addition" of reals is commutative and so that means that any of examples 2-5 which are groups will be abelian groups.