Download PDF by Benjamin Klopsch: Lectures on Profinite Topics in Group Theory
By Benjamin Klopsch
'In this e-book, 3 authors introduce readers to powerful approximation equipment, analytic pro-p teams and zeta features of teams. each one bankruptcy illustrates connections among countless staff thought, quantity thought and Lie thought. the 1st introduces the idea of compact p-adic Lie teams. the second one explains how tools from linear algebraic teams may be utilised to check the finite pictures of linear teams. Derived from an LMS/EPSRC brief path for graduate scholars, this booklet offers a concise advent to a really lively study sector and assumes much less earlier wisdom than latest monographs or unique learn articles. obtainable to starting graduate scholars in staff idea, it's going to additionally entice researchers drawn to countless workforce conception and its interface with Lie thought and quantity theory.'
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Extra info for Lectures on Profinite Topics in Group Theory
In the following, we denote the Zp -Lie lattice associated to G by L(G). The next proposition assures us that the assignment of a Lie lattice to a uniform pro-p group is well behaved with respect to the passage to subgroups or quotients. 7. Let G be a uniform pro-p group. Let H ≤c G be a uniform subgroup, and let N c G such that G/N is uniform. Then N is uniform and: (1) L(H) constitutes a Lie sublattice of L(G); (2) L(N ) constitutes a Lie ideal of L(G), the sets G/N and L(G)/L(N ) are equal and the natural epimorphism G → G/N of groups induces an epimorphism L(G) → L(G/N ) of Zp -Lie lattices with kernel L(N ).
4. Let G be a powerful ﬁnite p-group. Then rk(G) = d(G), in other words d(H) ≤ d(G) for all H ≤ G. Proof (by induction on |G|). Let H ≤ G and put d := d(G). Write Gi := Pi (G) for i ∈ N, and put d2 := d(G2 ). 2 shows that G2 is powerful, hence by induction the group K := H ∩ G2 satisﬁes d(K) ≤ d2 . Put e := d(HG2 /G2 ) so that e ≤ d. Our aim is to ﬁnd h1 , . . , he ∈ H and y1 , . . , yd−e ∈ K such that HG2 = h1 , . . , he G2 and K = hp1 , . . , hpe , y1 , . . , yd−e . This will imply H = h1 , .
11 (Pro-p groups of ﬁnite rank – summary of characterisations). Let G be a pro-p group. Then each of the following conditions is necessary and suﬃcient for G to have ﬁnite rank: (1) G is ﬁnitely generated and virtually powerful; (2) there exists r ∈ N such that every open subgroup of G contains an open normal subgroup N o G with d(N ) ≤ r; (3) G has polynomial subgroup growth; (4) G is isomorphic to a closed subgroup of GLd (Zp ) for suitable d ∈ N; (5) G is a p-adic Lie group. We conclude this section by stating an intriguing problem which aims at yet another interesting characterisation of pro-p groups of ﬁnite rank.
Lectures on Profinite Topics in Group Theory by Benjamin Klopsch