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By Mazurov V.D.

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4. (Stronger version) If G has a planar 2-cocycle with coefficients in a G-module A, then G and A are elementary abelian groups. 52 GUIDO’S BOOK OF CONJECTURES 23. 1. Let π be a finite group, and let Bπp∧ denote its pcompleted classifying space. , there exists an integer r ( depending on π ) such that pr · π∗ ((Bπ)∧p )) = {0}. Since π is finite, the fundamental group of Bπp∧ is the finite p-group given by the quotient of G by its minimal normal subgroup of p-power index, denoted by Op (π). If the order of Op (π) is not divisible by p, then Bπp∧ is homotopy equivalent to B(π/Op (π)), and the conjecture reduces to a triviality.

2. The number of relators needed to present Γn on the generators Sn goes to infinity as n → ∞, so Γn has a relation gap for n sufficiently large. Some virtually free examples. We consider groups similar in spirit to ones considered by D. Epstein, C. Hog-Angeloni, W. Metzler and M. Lustig, and more recently by K. Gruenberg and P. Linnell. Given letters xm and tm , let ρm be the word −1 −1 −m ρm = (tm xm t−1 m )xm (tm xm tm )xm . We look at the groups Γm,n = Qm ∗ Qn , where Qm = xm , tm ρm , xm−1 m and (mm−1 − 1) and (nn−1 − 1) are coprime.

A few are given by π = A5 , A6 , A7 , J4 , M11 at p = 2. A few more at the prime 2 are given by those finite simple groups of 2-rank 2 (including M11 ) with the possible exception of U (3, F4 ). The finite simple groups of classical Lie type over the field Fq , where q is a the power of a prime different from p provide a large family of examples at the prime p. GUIDO’S BOOK OF CONJECTURES 53 Finite simple groups of Lie type at the defining characteristic: Almost no examples are known of the behavior of πi (BG(Fpk )∧p ), where G is a finite simple groups of Lie type.

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A Characterization of Alternating Groups II by Mazurov V.D.

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