Consider the following system
It's clear that the Galilean transformation from the frame S to S' in this case is given by:
$$\begin{align} x' &= x-vt \\ y' &= y \\ z' &= z \\ t' &= t \end{align}$$
I'm asked to justify the second and third equations above using a symmetry argument. But I don't really see how the equations are even derivable -- they just seem like the obvious equations describing the system to me.
The only thing I can think is to say that, because $S(t=0) = S'(t'=0)$ and because the motion of $S'$ relative to $S$ is just a movement at the constant speed $v$ in the $x$-direction, there's no displacement in the $y$ or $z$ direction and hence $y(t=0)=y'(t'=0) \implies y(t)=y'(t')$ for all time $t = t'$. And likewise for $z$. But I don't think that counts as a symmetry argument (or does it? I'm not totally sure what a "symmetry" argument means -- it usually means use cylindrical or spherical symmetry to reduce the number of coordinates/ equations describing a system). What type of argument is being asked for here?
BTW, this is exercise 1 from page 45 of Resnick's Introduction to Special Relativity.
Symmetry often means just "identifying the structure in common between different things". I'm not sure this is what they want, given that I also feel "it's obvious" -- but it's one answer you could give: consider the case where $v = 0$. What happens to the equation for $x$? Now consider what must happen when I compose transformations, going once $+v$ then $-v$. Using the $x$ coordinate equation, we see that $x$ must be unchanged afterwards, which makes sense we're in the same frame as before. Similarly $y$ and $z$ must be unchanged.
But what if we go $+v$, then flip our definition of $x$ to $-x$, and then go $+v$ again? We return to the rest frame. But $y$ and $z$ can't be affected by just flipping our definition of $x$. So they must have experienced whatever transformation goes with $+2v$. Since they're unchanged, it stands that any boost in the $x$ direction can't affect $y$ or $z$.
Note: The slight caveat with this argument is that requires that physics has mirror symmetry. For most physical processes this is true, but in some cases it is not; notably weak quantum mechanics. Or some physical processes with a chiral medium, such as discussing optics in a water-dextrose solution. You could then theoretically have something where boosting in the $x$ direction causes $y$ and $z$ to rotate slightly clockwise. When we flip $x$ and boost backwards, it causes $y$ and $z$ to rotate counterclockwise back again, since they're sensitive to the left- or right-handedness of the coordinate system.