Are my answers on the right track?
How many symmetries does a square have? Describe them.
8 symmetries: 4 reflexives, 3 rotational (90, 180, and 270), 1 trivial
Working in the "symmetry group" of a square, how many solutions does the equation $x^2=e$ have? What about the equation $x^4=e$?
For $x^2 = e$ I believe there are 6 solutions: 4 reflexives, 1 rotational, 1 trivial
I am confused for $x^4=e$.
How many symmetries does a regular n-gon have? (This means an n-sided polygon, all of whose sides are equal and all of whose angles are equal.)
Ans: $2n$ (line of symmetry, and angles )
Working in the symmetry group of the regular n-gon, how many solutions does the equation $x^2=e$ have? What about the equation $x^n=e?$ (Warning: the answers depend on whether n is even or odd.)
For the equation $x^2=e$, is it just $n$ for line of symmetries?
I am confused by the $x^n=e$.
If $\rho$ is one of the rotations, what is $\rho^4$? Also, if something satisfies $x^2 = e$ it must also satisfy $x^4 = e$.
Now we can try to bump this up to $n >4$. If $n$ is even and $\rho$ is a rotation then what is $\rho^n$? Since $n$ is even, anything that satisfies $x^2=e$ also satisfies $x^n=e$. How does this change when $n$ is odd?