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Description: In this note we give definitions concerning free groups for forms [1 BB], [1 CA], Form 68 and Form 348.

Content:

In this note we give definitions concerning free groups for forms [1 BB], [1 CA], Form 68 and Form 348.

Definition: Assume that $(G,\circ)$ is a group freely generated by $X$, that $A\subseteq G$, and that $a$, $b$ and $c$ are in $G$.

1. The subgroup of $G$ generated by $A$ is denoted $\langle A \rangle$.
2. The length of $a$ with respect to $X$ (denoted by $L_X(a$ or $L(a)$) is the unique natural number $n$ such that there are elements $a_1,\ldots,a_n$ of $X\cup X^{-1}$ such that $a_j \ne a_{j+1}^{-1}$ and $a = a_1\circ\cdots\circ a_n$.
3. The subset $A$ is level with respect to $X$ if $\langle A\rangle$ is freely generated by $A$ and for each $b\in\langle A\rangle$, $b\in \langle \{a : a\in A \land L(a)\le L(b)\}$.
4. A subset $A$ of $G$ has the Nielsen property with respect to $X$ if
   1. $A\cap A^{-1} = \emptyset$.
   2. If $a,b\in A\cup A^{-1}$ and $L(a\circ b) < L(a)$ then $b = a^{-1}$, and
   3. If $a$, $b$ and $c \in A\cup A^{-1}$ and $L(abc)\le L(a) - L(b) + L(c)$ then either $b = a^{-1}$ or $c = b^{-1}$.

In Federer/Jonsson [1950] it is shown that a set with the Nielsen property is level.

Howard-Rubin number: 129

Type: Definitions

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