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 \(G\).
- The subgroup of \(G\) generated by \(A\) is denoted \(\langle A \rangle\).
- 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
\(a = a_1\circ\cdots\circ a_n\).
- The subset \(A\) is level with respect to \(X\) if
\(\langle A\rangle\) is freely generated by \(A\) and for each
\(b\in\langle A\rangle\), \(b\in \langle \{a : a\in A \land L(a)\le
L(b)\}\).
- A subset \(A\) of \(G\) has the Nielsen property with
respect to \(X\) if
- \(A\cap A^{-1} = \emptyset\).
- If \(a,b\in A\cup A^{-1}\) and \(L(a\circ b) < L(a)\) then
\(b = a^{-1}\), and
- 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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