# 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 Cayleyâs theorem#

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

Because \(p_i Â· p_j = p_k\)

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 authorâs footnote regarding composition is a bit unfortunate. Itâs fine and good to choose this particular convention of right-multiplication for composition: itâs also the choice of GAP, Permutations - SymPy and Permutations Â· AbstractAlgebra.jl. Whatâs unfortunate about this choice is that (unlike the mentioned libraries) the author chooses to use the symbol â (âring operatorâ in Unicode) for composition.

Per Function composition Â§ Alternative notations, the more common notational preference is \(fâg\) for âg then fâ although it gives mention to the authorâs convention. Weâll prefer \(fâšg\) for âf then gâ when it doesnât conflict with the author.

Although the authorâs right-multiplication is preferred going forward, notice Permutation Â§ 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.