A field guide to groups
A group is a set of moves you can combine and undo. There are exactly two of order 4, five of order 8, and one of order 15. The small ones turn up everywhere, and most of them are easy to tell apart once you know what to look for.
Each plate below is the group's full multiplication table, drawn in colour. Every element gets its own colour, and the square in row $x$, column $y$ is coloured by the product $xy$.
This is $S_3$, the six ways to rearrange three things. The grey squares are the identity. Every row and every column contains every colour exactly once. That's true of every group, and it's the reason you can always undo a move.
Click a plate to switch to the commuting view, which lights up the pairs with $xy = yx$. An abelian group lights up completely.
What the labels on each card mean
- order
- How many elements the group has.
- abelian
- Whether every pair commutes: xy = yx, so the order you combine things in never matters.
- centre
- The elements that commute with everything in the group. The identity always does.
- classes
- Conjugacy classes: bundles of elements that are the same move seen from a different angle, like the three flips of a triangle.
- commuting
- How many ordered pairs (x, y) have xy = yx. For a non-abelian group it is never more than 5/8 of them.
- element orders
- For each number n along the bottom, how many elements take exactly n repeats to get back to the identity. It is the best single fingerprint of a small group.
- field marks
- What to look for to recognise it.
- habitat
- Where it turns up in maths and in the world.
- mistaken for
- The look-alike it gets confused with, and how to tell them apart.
Identify a specimen
Give some permutations in cycle notation. The key generates the group they make and works out which species it is. Products are read right to left, like functions.