e-book Biset functors for finite groups

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Cancel Submit. This book is intended both to students and researchers, as it gives a didactic exposition of the basics and a rewriting of advanced results in the area, with some new ideas and proofs. Review From the reviews: "The theory of biset functors was developed to give a unified construction of the elementary operations of restriction, induction, inflation, deflation and transport by isomorphism. This volume gives an excellent introduction to the topic, which until now was only available in a series of research papers.

The Roquette category TZp is an additive tensor category. Let P he a finite p-group, and B be a genetic basis of P. Let P be a finite p-group, and B be a genetic basis of P. On the other hand, by Proposition 6. See For Assertion 3, observe that by Lemma [2. Corollary : Let P he a finite p- group.

Let A, B, and C he the subgroups of index 2 in P. Then P has a unique genetic basis, consisting of P, and all its subgroups of index p. The tensor structure 4. Notation : Let G and H be groups. Definition : Let G and H he groups. Notation : Let G and H he groups. Theorem : Let P and Q be non-trivial Roquette p-groups, let cp resp. Zq denote the central subgroup of order p in P resp. Then L is a centrally diagonal genetic subgroup of PxQ. For any x,? In particular, for?

Hence 4. In particular 4.

Finite Groups...

This proves part a of Assertion 2. In particular H is cyclic, and non-trivial. This completes the proof of Assertion 1. Let K denote an axial subgroup of Q of order ep. Similarly, the map TTK-. Moreover 4. It is a Roquette group, independent of ip, up to isomorphism. So Im a only depends on the type of P. Case 2 : Suppose now that IpQ has order 2, i. In the exact sequence This completes the proof of Theorem I4. Theorem : Let P and Q be Roquette p-groups, of exponents cp and cq, respectively.

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These conditions depend only on the double coset Npy. Hence This completes the proof in this case. Let H and H' be subgroups of order 4 of P. The normalizer of L in P x Q does not depend on ip, by I4. Let L and L' be two such centrally diagonal genetic subgroups of P x Q. Corollary : Lei P and Q be Roquette p-groups. By Theorem I3. Corollary SSI follows from the fact that 4.

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In each case the sum This completes the proof. Examples and applications 5. Suppose first that p is odd. The "multiplication rule" of the edges dCpn is the following 5. Proposition : In TZ2, the edge dC2 is isomorphic to the trivial group 1 or its edge dl. Let X, Y and Z denote the subgroups of order 2 oi E.

So this morphism is equal to 0.

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It follows that 5. So a and b are mutual inverse isomorphisms between 1 and dX. Corollary : Let F be a rational 2-biset functor. Proof : Indeed rational p-biset functors are exactly those p-biset functors which factor through the category Tip. The second one follows from Example Remark : The "action" of this involution on the edges of the other Roquette 2-groups that is, different from 1, C2, and Qs is as follows : it stabilizes cyclic and semidihedral groups, and exchanges dihedral and gen- eralized quaternion groups.

More precisely, it follows from Corollary By Theorem Proposition: Let S and T be finite sequences of Roquette p-groups, such that there exists an isomorphism 5. It follows that for any Roquette p-group R, the number of terms in the sequence S which are isomorphic to R is equal to the corresponding number in the sequence T. The proposition follows in this case. The isomorphism Now, let C be a primitive i. Applying this functor to the isomorphism Proposition : Let P and Q be finite p-groups.

The following assertions are equivalent : 1. The groups P and Q are isomorphic in the category TZp. Proof : Assertion 2 implies Assertion 1 by Theorem Let Bp and Bq be genetic bases of P and Q, respectively.

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Assertion 2 follows. The equivalence of Assertions 2 and 3 follows from Theorem I2. In this case however, the following result characterizes those p-groups which become isomorphic in the category Tip : 5. Proposition : Let p be a prime number, and let P and Q be finite p- groups. This proves Assertion 1. By Exam- ple By Galois theory [15] Theorem 1.

Then the G-set B.

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Now Assertion 2 of the proposition follows from Proposition Remark : Assertion 2 of Proposition By Example Genetic bases of direct products.