# 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 Cayleyâs theorem#

Compare Cayleyâs theorem, which provides a little cleaner proof. That article defines $$f_g(x) = gx$$ i.e. using left-multiplication rather than the authorâs right multiplication $$p_k(x) = xk$$; the authorâs choice of composition symbol is unfortunate but choosing the opposite convention is not a problem (see more below). Whatâs strange about the authorâs 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.

Per Function composition Â§ Alternative notations, the more common notational preference is $$fâg$$ for âg then fâ although it gives mention to the authorâs convention. Weâll prefer $$fâšg$$ for âf then gâ when it doesnât conflict with the author.