5 Five families

5.1 Cyclic groups

The author introduces the concept of a Cycle graph (algebra) in this section; distinguish from Cycle graph.

5.4 Symmetric and alternating groups

Compare Alternating group.

5.4.4 Cayley’s theorem

Compare Cayley’s theorem, which provides a little cleaner proof. That article defines \(f_g(x) = gx\) i.e. using left-multiplication rather than the author’s right multiplication \(p_k(x) = xk\); the author’s choice of composition symbol is unfortunate but choosing the opposite convention is not a problem (see more below). What’s strange about the author’s proof is its focus on how permutations act on the identity, and that it simply uses:

Because \(p_i · p_j = p_k\)

Without really establishing that this will hold. That is, the author essentially assumes the permutations will form a permutation group by assuming they will form a multiplication table. Replacing \(1\) with \(x\) in the proof would make it clearer that we still have a group.

See also Yoneda lemma.

Composition

The author’s footnote regarding composition is a bit unfortunate. It’s fine and good to choose this particular convention of right-multiplication for composition: it’s also the choice of GAP, Permutations - SymPy and Permutations · AbstractAlgebra.jl. What’s unfortunate about this choice is that (unlike the mentioned libraries) the author chooses to use the symbol ∘ (“ring operator” in Unicode) for composition.

Per Function composition § Alternative notations, the more common notational preference is \(f∘g\) for “g then f” although it gives mention to the author’s convention. We’ll prefer \(f⨟g\) for “f then g” when it doesn’t conflict with the author.

Although the author’s right-multiplication is preferred going forward, notice Permutation § Composition of permutations also uses the opposite convention. It unfortunately also often takes the same liberty as our author of not using a symbol between the elements (concatenation) to disambiguate.