What are the left and right cosets of D₄?
I know that there are 24 elements in S₄ and 8 in D₄, which gives 24÷8=3 distinct left and right cosets.
I'm having trouble multiplying permutations and have no idea if I'm doing them correctly.
For example, (1234)(1234)=13, (1234)(13)(24)=(1432), (1234)(1432)=e
Are any of these even right?
Best Answer
"What are the left and right cosets of $D_4$?" does not make sense. You have to ask "what are the left and right cosets of H in G?", where H is some subgroup of G. NasuSama's answer has given you a good way of calculating the cosets of $\{e, y^2\}$ in $D_4$, but I suspect you wanted to know the cosets of $D_4$ in $S_4$.
First of all, cycle notation. You should think about $(1234)$ as a function, let's call it $f$, such that
Similarly, $(13)(24)$ is a function, let's call it $g$, such that
Do you see why? Now, when you "multiply" f by g, you simply find the function $fg$ (or $f\circ g$ if you prefer). So $fg(1) = f(g(1)) = f(3) = 4$, and so on. (Warning: some people write $fg$ to mean $gf$. Both notations are in use - check with your lecturer or textbook.)
Do you now understand how to form cosets? To find the left coset of $D_4$ in $S_4$ corresponding to the element $(123)$, just left-multiply everything in $D_4$ by $(123)$.
Here are a few helpful facts about cosets of $H$ in $G$:
Everything I said above works with left cosets replaced by right cosets. Be careful when you mix left with right, though: