Does the set $\{1, i , −1\}$ form a group? 
Does the set $\{1, i , −1\}$ form a group?

This is exactly what the question in my book asks.
I am not sure how to answer this, I would wait it says to me if the group is defined under ordinary multiplication or in another way. Of course if it is the former is no, but what about the latter?
Am i missing something?
 A: It does not make grammatical sense, mathematically, to ask whether a set is a group, precisely because no operation has been specified. Many people do this and many people are being sloppy. The question that makes sense is to ask whether a subset of a group is a subgroup; in this case $\{ 1, i, -1 \}$ is a subset of, say, $\mathbb{C}^{\times}$, which is not a subgroup because it's not closed under (ordinary) multiplication.
Another question that makes sense is to equip a set $S$ with a binary operation $\ast : S \times S \to S$ (giving a magma) and ask whether that operation satisfies the group axioms. But again no operation has been specified here. A third question that makes sense is to ask whether a set $S$ admits some group operation, but obviously every finite set has this property, and in fact every set does (this is equivalent to the axiom of choice).
Most likely the intended question is whether $\{ 1, i, -1 \}$ is a subgroup of $\mathbb{C}^{\times}$.
A: No (and yes${}^\dagger$).
No operation is defined for it, so it makes no sense to ask.
Assuming multiplication, since $-1\times i=-i\notin\{1,i,-1\}$, it is not closed, so it cannot be a group.

$\dagger$: If you want, you could pick any element of $\{1,i,-1\}$ to be the identity element $e$ with respect to concatenation, one of the remaining two elements to be the square of the other, call them $a$ and $a^2$, with $a^3=e$, so that, under this operation, the set is a group (isomorphic to $\Bbb Z_3$).
A: The 4-element set $\{1,-1,i,-i\}$ does form a group under multiplication--indeed they are the quartic roots of 1, and the 3 elements in $\{1,i, -1\}$ generate that 4-element group.
But I agree with the others--sloppily written. And perhaps a typo too, e.g., the author meant to include $-i$ as well?
