Exercise 25, § 4

Let E be a set and $f$ a mapping of $E^m$ into E; we write

suppose that $f$ satisfies the following conditions:

(1)

identically; (2) For all $a_1, a_2, \ldots, a_{m-1}$, the mappings

are bijections of E onto E.

(a) Show that identically

for every index $i$ such that $1 \leq i \leq m-1$ (argue by induction on $i$, by considering the element

(b) For all $a_1, a_2, \ldots, a_{m-2}$, there exists $u \in E$ such that for all $x$

(c) In the set $E^k$ of sequences $(u_1, u_2, \ldots, u_k)$ of $k$ elements of $E$ ($1 \leq k \leq m - 1$) consider the equivalence relation $R_k$ which states: for all $x_1, x_2, \ldots, x_{m-k}, u_1u_2\ldots u_kx_1x_2\ldots x_{m-k} = v_1v_2\ldots v_kx_1x_2\ldots x_{m-k}$; let $E_k$ denote the quotient set $E^k / R_k$ and $G$ the sum set (Set Theory, II, § 4, no. 8) of $E_1 = E, E_2, \ldots, E_{m-1}$. Let $\alpha \in E_i, \beta \in E_j$; if $(u_1, u_2, \ldots, u_i)$ is a sequence of class $\alpha$, $(v_1, \ldots, v_j)$ a sequence of class $\beta$, consider the sequence

in $E^{i+j}$ if $i + j < m$, the sequence in $E^{i+j-m+1}$ if $i + j \geq m$; show that the class of this sequence in $E_{i+j}$ (resp. $E_{i+j-m+1}$) depends only on the classes $\alpha$ and $\beta$; it is denoted by $\alpha . \beta$. Show that a group law is thus defined on $G$, that $H = E_{m-1}$ is a normal subgroup of $G$ and that the quotient group $G/H$ is cyclic of order $m - 1$; prove finally that $E$ is identical with a class (mod. $H$) which generates $G/H$ and that $x_1 x_2 \ldots x_m$ is just the composition, in the group $G$, of the sequence $(x_1, x_2, \ldots, x_m)$.