Fee subgroup
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A fee subgroup \(H\) of a linearly ordered subgroup \(G\)is a subgroup such that \(g^{-1}Hg \subset H\) for all \(g > 1\).
When a map \(a\) defined on a linearly ordered group \(G\) satisfies that \(a(g) = a(h)\) implies \(a(fg) = a(fh)\) for all \(f > 1\), \(P(a) = \{g^{-1}h \mid a(g) = a(h)\}\) is a fee subgroup[1].
A fee subgroup is not always normal [2].