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The dihedral group of symmetries of the square, $D_8$, is given by

$$D_8 = \{e, r, r^2, r^3, s, sr, sr^2, sr^3\}$$

where $e$ is the identity and the generators $r$ and $s$ satisfy

$$r^4=e, s^2 = e, r^is=sr^{4-i}, i=1,2,3$$

$a$) State the order of each element of $D_8$.

Really not sure where to start with this question if i'm honest.

$b$) Show that the subset $N = \{e,r^2\}$ is a subgroup of $D_8$

To show that $N$ is a subgroup we need to show $N$ is non empty and $a,b^{-1}\in N, \forall a,b \in N$

First one is satisfied, not sure how to compute the second part.

$c)$ Show that $N$ is normal in $D_8$

In my solution to this I have shown that left and right cosets are equal.

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Hints:

a) How many times do you need to multiply $r$ by itself to get $e$? This tells you the order of $r$. Do the same thing for the other elements. You will need to use the three relations. Or you can think about how many times you need to successively apply each rotation/reflection to get the identity.

b) $(r^2)^2=e$ so $N$ is closed and contains inverses of its elements since $(r^2)^{-1}=r^2$. It also contains the identity, so it is a subgroup.

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I'm thinking $1,4,2,4,2,2,2,2$, respectively. Just investigate the symmetries. (Check me on this.)

For normality, $e$ clearly commutes with everything. Then for $r^2$, it clearly commutes with powers of $r$. As for $sr^2=r^2s$ by the relation given. This should do it.

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