Sets/Groups and inverse? "If $x * y = 5xy$ then the inverse of $2$ w.r.t to the group operation $*$ is ____?" 
I have no idea on how to go about doing this question.... does it mean a multiplicative inverse of that equation or something? 
Can somebody please explain this to me like I'm a complete idiot. The q is a MCQ and the final answer is $1/50$.
 A: Your goal in this problem is to simply substitute in definitions, with the goal of converting the given question to one or more algebra problems.
For example, the definition of "inverse of $2$ with respect to $*$" is

$x$ is the inverse of $2$ with respect to $*$ if and only if $2 * x = i$ and $x * 2 = i$, where $i$ the identity for $*$

Similarly, when you see the equation $2 * x = i$, you can simply substitute in the definition of $*$ to get $5 \cdot 2 \cdot x = i$, and similarly for other equations.
Note that you need to know what the identity is. That's a separate problem you can solve: i.e. you want to solve the following exercise:

Find the identity element for $*$

A: Recall that an element $a$ is called inverse of $b$ (with respect to the operation $\ast$) when $a\ast b=b \ast a=e$ where $e$ is the neutral element.  
Thus first of all you need to figure out what $e$ is or if it even exists. 
Is there an element $e$ such that $x\ast e = e\ast x = x$ for every $x$. 
You'll find that yes there is one and it is $1/5$ because $x\ast 1/5 = 5 x (1/5)=x$. 
Having found that it remains to solve $2  \ast x = 1/5$ that is $5(2 x) = 1/5  $, which is your result. 
A: Hint: Find the identity element with respect to $*$ first. Then you will be able to find the inverse easily. 
