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Factor Groups and Subgroups
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amalgamated associative algebra automorphisms B. H. Neumann basic Lie elements commutator subgroup conjugate Corollary coset cyclic group cyclically reduced deﬁnes an element deﬁnition derivable element of G elements of degree elements of ﬁnite empty word exponent sum extended Schreier system ﬁnd ﬁnite group ﬁnite index ﬁnite order ﬁnitely presented ﬁrst ﬁxed follows free Abelian group free factor free group free product freely equal freely reduced word G mod H given graph group G H int Hence homomorphism Hopﬁan initial segment integer inverse isomorphic Lemma Let F Let G let H Lie algebra lower central series mapping Math matrix module monomial Moreover multiplication Nielsen transformation nilpotent normal subgroup obtain polynomial quotient group replaced representative system residually ﬁnite result rewriting process s-symbols satisﬁes sequence single deﬁning relator solve the word subgroup H subgroup of G Suppose syllable length symbols Tietze transformations verbal subgroup W(av word problem zero