TITLE: Interpretation of Symbol: "$\rtimes$"
QUESTION [5 upvotes]: What exactly does $\text{Aff}(2) = \mathbb{R}^2 \rtimes SL_{2}(\mathbb{R})$ mean? I know that it is the group of area preserving affine transformations of (oriented) $\mathbb{R}^2$. But how would you interpret the $\rtimes$ symbol?

REPLY [4 votes]: The hardest thing about the notation $N \rtimes H$ is remembering which of the groups is the normal subgroup and which is the group acting.  There are two good mnemonics for this.  The first is that $\rtimes$ contains the symbol $\vartriangleleft$ for a normal subgroup and the normal subgroup is on the correct side of the symbol.  The other mnemonic is to hold your arms in front of you and move them up and down.  In profile you will make a $\rtimes$ and then you just remember that you are acting on something so the hands go on the side with $N$ which is being acted on while $H$ goes on the side that you're on doing the acting.