By Wilhelm Magnus
This seminal, much-cited account starts with a pretty user-friendly exposition of simple innovations and a dialogue of issue teams and subgroups. the themes of Nielsen variations, loose and amalgamated items, and commutator calculus obtain specified therapy. The concluding bankruptcy surveys be aware, conjugacy, and comparable difficulties; adjunction and embedding difficulties; and extra. moment, revised 1976 version.
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Extra resources for Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations (Dover Books on Mathematics)
Sn ⊆ U , and suppose that we are interested in reconstructing S1 , . . , Sn , or in ﬁnding k elements of each Si (for some parameter k). We are provided with procedures such that for any set A ⊆ U , we can compute: – Intersection Cardinality: ISize(S1 , . . , Sn , A) = |S1 ∩ A|, . . , |Sn ∩ A| . – Intersection Sum: ISum(S1 , . . , Sn , A) = u∈S1 ∩A u, . . , u∈Sn ∩A u . Clearly, given a suﬃcient number of calls to these procedures, it is possible to fully reconstruct S1 , . . , Sn . However, we aim at doing so with a minimal amount of work.
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Recall that given a collection S containing n subsets of U , the k-reconstruction problem is: For each S ∈ S, ﬁnd min(k, |S|) elements of S. The bounded k-reconstruction problem is: Fully reconstruct every set S ∈ S of size at most k. For the bounded k-reconstruction problem, bounded k-peelers fully solve the problem. Thus, we obtain: Theorem 4. Suppose that computing ISize(S, A) or ISum(S, A) for all S ∈ S and one set A takes O(f ) steps. Then, there exist deterministic and randomized algorithms for the bounded k-reconstruction problem with the following running times: – Deterministic: O(k · polylog(m)(f + n)) steps.